❲■▲▲■❆▼❙ ❯❇❆▲❉❖ ❍❯❆▼❆◆■ ◗❯■❙P❊

❯▼ ❏❖●❖ ■◆❋■◆■❚❖ ❈❖▼ ❈❖◆❙❊◗❯✃◆❈■❆❙ ❚❖P❖▲Ó●■❈❆❙

❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ à ❯♥✐✈❡rs✐❞❛❞❡❋❡❞❡r❛❧ ❞❡ ❱✐ç♦s❛✱ ❝♦♠♦ ♣❛rt❡ ❞❛s ❡①✐✲❣ê♥❝✐❛s ❞♦ Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦❡♠ ▼❛t❡♠át✐❝❛✱ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦❞❡ ▼❛❣✐st❡r ❙❝✐❡♥t✐❛❡✳

❱■➬❖❙❆▼■◆❆❙ ●❊❘❆■❙ ✲ ❇❘❆❙■▲

✷✵✶✼

❉❡❞✐❝♦ ❡st❡ tr❛❜❛❧❤♦ ❛ ♠✐♥❤❛ ❢❛♠í❧✐❛✳

✐✐

▲♦ q✉❡ ❞✐st✐♥❣✉❡ ❧♦ r❡❛❧ ❞❡ ❧♦

✐rr❡❛❧ ❡stá ❡♥ ❡❧ ❝♦r❛③ó♥

❏♦❤♥ ◆❛s❤✳

✐✐✐

❆❣r❛❞❡❝✐♠❡♥t♦s

Pr✐♠❡✐r❛♠❡♥t❡ ❡✉ ❛❣r❛❞❡ç♦ ❛ ❉❡✉s ♣♦r ♠❡ ❛❜❡♥ç♦❛r✱ ✐❧✉♠✐♥❛r ♥♦s ♠♦♠❡♥t♦s♠❛✐s ❞✐❢í❝❡✐s✱ ❡ ♠❡ ❣✉✐❛r s❡♠♣r❡✳

❆❣r❛❞❡ç♦ ❛ ♠✐♥❤❛ ❢❛♠í❧✐❛✱ ♣♦r ♠❡ ♠♦t✐✈❛r✱ ❛❝r❡❞✐t❛r ❡♠ ♠✐♥ ❡ t♦r❝❡r ♣♦r ♠✐♥✳❖❜r✐❣❛❞♦ ♣♦r s❡r ♠✐♥❤❛ ♠♦t✐✈❛çã♦ ✦✳

❆❣r❛❞❡ç♦ ❛ ♠✐♥❤❛ ♦r✐❡♥t❛❞♦r❛✱ ❈❛t❛r✐♥❛✱ ♣❡❧❛ ♣❛❝✐ê♥❝✐❛✱ ❛♣r❡♥❞✐③❛❞♦ ✈❛❧✐♦s♦✱♣♦r ❝♦♥✜❛r ❡♠ ♠✐♥✱ ♣❡❧❛s s✉❛s ❝♦rr❡çõ❡s ❡ ✐♥❝❡♥t✐✈♦✳ ❊♥✜♠ ♣❡❧❛ ♣❡ss♦❛♠❛r❛✈✐❧❤♦s❛ q✉❡ é✳

❆❣r❛❞❡ç♦ ❛ ♠❡✉s ❝♦❧❡❣❛s ❞❡ ❝✉rs♦ ♣❡❧❛ ❛♠✐③❛❞❡ ❡ ❛ t♦❞♦s q✉❡ ❞❡ ❛❧❣✉♠❛ ❢♦r♠❛❝♦♥tr✐❜✉ír❛♠ ♣❛r❛ ❛ r❡❛❧✐③❛çã♦ ❞❡st❡ tr❛❜❛❧❤♦✳

❆♦s ♣r♦❢❡ss♦r❡s ❡ ❢✉♥❝✐♦♥ár✐♦s ❞♦ ❉▼❆✲❯❋❱✱ ♣♦r ❝♦❧❛❜♦r❛r❡♠ ❝♦♠ ❛ ♠✐♥❤❛❢♦r♠❛çã♦ ❡ ♣❡❧♦s ❡✜❝✐❡♥t❡s s❡r✈✐ç♦s ♣r❡st❛❞♦s✳

❋✐♥❛❧♠❡♥t❡✱ ❛❣r❛❞❡ç♦ à ❈❆P❊❙ ❡ ❋❆P❊▼■● ♣❡❧♦ ❛♣♦✐♦ ✜♥❛♥❝❡✐r♦✐♥❞✐s♣❡♥sá✈❡❧ ♣❛r❛ ❛ r❡❛❧✐③❛çã♦ ❞❡st❡ tr❛❜❛❧❤♦✳

✐✈

❙✉♠ár✐♦

❘❡s✉♠♦ ✈✐✐

❆❜str❛❝t ✈✐✐✐

■♥tr♦❞✉çã♦ ✶

✶ Pr❡❧✐♠✐♥❛r❡s ✸

✶✳✶ ❚♦♣♦❧♦❣✐❛ ❡ ❇❛s❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸

✶✳✷ ❚♦♣♦❧♦❣✐❛ ■♥✐❝✐❛❧ ❡ Pr♦❞✉t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻

✶✳✸ ❈♦❜❡rt✉r❛s ❞❡ ❊s♣❛ç♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾

✷ ❊s♣❛ç♦s ❯♥✐❢♦r♠❡s ✶✷

✷✳✶ ❯♥✐❢♦r♠✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷

✷✳✷ ❇❛s❡ ❯♥✐❢♦r♠✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺

✷✳✸ Pr♦♣r✐❡❞❛❞❡s ❞❡ ❯♥✐❢♦r♠✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽

✸ ❏♦❣♦ ❚♦♣♦❧ó❣✐❝♦ Pr♦①✐♠❛❧ ✷✷

✸✳✶ ❏♦❣♦ Pr♦①✐♠❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷

✸✳✷ ❊s♣❛ç♦s Pr♦①✐♠❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺

✸✳✸ Pr♦♣r✐❡❞❛❞❡s ❞❡ ❊s♣❛ç♦s Pr♦①✐♠❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵

✹ ❏♦❣♦ ❚♦♣♦❧ó❣✐❝♦ ●r✉❡♥❤❛❣❡ ✸✹

✹✳✶ ❏♦❣♦ ●r✉❡♥❤❛❣❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹

✹✳✷ W ✲ ❊s♣❛ç♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻

✹✳✸ Pr♦♣r✐❡❞❛❞❡s ❞❡ W ✲❊s♣❛ç♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽

✈

✺ ❈♦♥s❡q✉ê♥❝✐❛s ❚♦♣♦❧ó❣✐❝❛s ✹✶

✺✳✶ ❏♦❣♦ Pr♦①✐♠❛❧ ❋r❛❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶

✺✳✷ ❙❡♣❛r❛çã♦ ❡ ❈♦❜❡rt✉r❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺

✺✳✸ ❈♦♥s❡q✉ê♥❝✐❛ ❚♦♣♦❧ó❣✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵

❈♦♥s✐❞❡r❛çõ❡s ❋✐♥❛✐s ✺✷

❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ✺✸

✈✐

❘❡s✉♠♦

❲■▲▲■❆▼❙✱ ❯❜❛❧❞♦ ❍✉❛♠❛♥✐ ◗✉✐s♣❡✱ ▼✳❙❝✳✱ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❱✐ç♦s❛✱❞❡③❡♠❜r♦ ❞❡ ✷✵✶✼✳ ❯♠ ❏♦❣♦ ■♥✜♥✐t♦ ❝♦♠ ❈♦♥s❡q✉ê♥❝✐❛s ❚♦♣♦❧ó❣✐❝❛s✳❖r✐❡♥t❛❞♦r❛✿ ❈❛t❛r✐♥❛ ▼❡♥❞❡s ❞❡ ❏❡s✉s ❙á♥❝❤❡③✳

◆♦ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ ❡st✉❞❛♠♦s ✉♠ ❥♦❣♦ t♦♣♦❧ó❣✐❝♦ ✐♥✜♥✐t♦ ❞❡✜♥✐❞♦ ♥✉♠

❡s♣❛ç♦ ✉♥✐❢♦r♠❡✱ q✉❡ ❣❡r❛ ✉♠ ♥♦✈♦ ❡s♣❛ç♦✱ ❝❤❛♠❛❞♦ ❡s♣❛ç♦ ♣r♦①✐♠❛❧✳ ▼❛✐s

❡s♣❡❝✐✜❝❛♠❡♥t❡✱ ❡st✉❞❛♠♦s ❛s ♣r♦♣r✐❡❞❛❞❡s ❞❡st❡ ❡s♣❛ç♦ ❝♦♠♦ s✉❛ r❡❧❛çã♦ ❝♦♠

W ✲❡s♣❛ç♦s✱ t❡♦r❡♠❛ ❞❡ ❝❛r❛❝t❡r✐③❛çã♦ ❞❡ ❡s♣❛ç♦ ♠étr✐❝♦ ❝♦♠♣❧❡t♦✱ ❡ ❛❧❣✉♠❛s

❝♦♥s❡q✉ê♥❝✐❛s t♦♣♦❧ó❣✐❝♦s✱ ❝♦♠♦ ✐♠♣❧✐❝❛çõ❡s ❛ ❡s♣❛ç♦s ❝♦❧❡çã♦ ♥♦r♠❛❧✱ ❡s♣❛ç♦

♠❡t❛✲❝♦♠♣❛❝t♦ ❡ ❝♦❧❡çã♦ ❍❛✉s❞♦r✛✳

✈✐✐

❆❜str❛❝t

❲■▲▲■❆▼❙✱ ❯❜❛❧❞♦ ❍✉❛♠❛♥✐ ◗✉✐s♣❡✱ ▼✳❙❝✳✱ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❱✐ç♦s❛✱❉❡❝❡♠❜❡r✱ ✷✵✶✼✳ ❆♥ ■♥✜♥✐t❡ ●❛♠❡ ✇✐t❤ ❚♦♣♦❧♦❣✐❝❛❧ ❈♦♥s❡q✉❡♥❝❡s✳❆❞✈✐s❡r✿ ❈❛t❛r✐♥❛ ▼❡♥❞❡s ❞❡ ❏❡s✉s ❙á♥❝❤❡③✳

■♥ t❤✐s ♣❛♣❡r ✇❡ st✉❞② ❛♥ ✐♥✜♥✐t❡ t♦♣♦❧♦❣✐❝❛❧ ❣❛♠❡ ❞❡✜♥❡❞ ✐♥ ❛ ✉♥✐❢♦r♠ s♣❛❝❡✱

t❤❛t ❣❡♥❡r❛t❡s ❛ ♥❡✇ s♣❛❝❡✱ ❝❛❧❧❡❞ ♣r♦①✐♠❛❧ s♣❛❝❡✳ ▼♦r❡ s♣❡❝✐✜❝❛❧❧②✱ ✇❡ st✉❞②

t❤❡ ♣r♦♣❡rt✐❡s ♦❢ t❤✐s s♣❛❝❡ ❛s ✐ts r❡❧❛t✐♦♥ ✇✐t❤ W ✲s♣❛❝❡✱ ❝♦♠♣❧❡t❡ ♠❡tr✐❝

s♣❛❝❡ ❝❤❛r❛❝t❡r✐③❛t✐♦♥ t❤❡♦r❡♠✱ ❛♥❞ s♦♠❡ t♦♣♦❧♦❣✐❝❛❧ ❝♦♥s❡q✉❡♥❝❡s✱ s✉❝❤ ❛s

✐♠♣❧✐❝❛t✐♦♥s ❢♦r ♥♦r♠❛❧ ❝♦❧❧❡❝t✐♦♥ s♣❛❝❡s✱ ♠❡t❛✲❝♦♠♣❛❝t s♣❛❝❡✱ ❛♥❞ ❍❛✉s❞♦r✛

❝♦❧❧❡❝t✐♦♥✳

✈✐✐✐

■♥tr♦❞✉çã♦

❆ ♥♦çã♦ ❞❡ ❥♦❣♦ t♦♣♦❧ó❣✐❝♦ t❡♠ ♥♦ ♣r✐♥❝✐♣✐♦ ❛ ❇❛♥❛❝❤ ❡ ▼❛③✉r ❡♠ ✶✾✸✺✱♦♥❞❡ s❡ ♣r♦♣ôs ♥♦ ❢❛♠♦s♦ ❧✐✈r♦ ❞❡ ♣r♦❜❧❡♠❛s ❞❡ ❇❛♥❛❝❤✱ ✉♠ ❥♦❣♦ q✉❡ ❡st❛✈❛r❡❧❛❝✐♦♥❛❞♦ ❝♦♠ ❛ ♣r♦♣r✐❡❞❛❞❡ ❞❡ ❇❛✐r❡ ❞❡ ✉♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦✳

❆ ✜❧♦s♦✜❛ ❣❡r❛❧ é ♠❛✐s ♦✉ ♠❡♥♦s ❛ s❡❣✉✐♥t❡✿ ❞❡✜♥❡✲s❡ ✉♠ ❥♦❣♦ ❡♥tr❡ ❞♦✐s❥♦❣❛❞♦r❡s ❆ ❡ ❇✱ ❡♥✈♦❧✈❡♥❞♦ ❛ t♦♣♦❧♦❣✐❛ ❞♦ ❡s♣❛ç♦ ❡♠ q✉❡stã♦ ❡✱ ♣r♦✈❛✲s❡ ♣♦r❡①❡♠♣❧♦✱ q✉❡ ✉♠ ❝❡rt♦ ❡s♣❛ç♦ t❡♠ ❝❡rt❛ ♣r♦♣r✐❡❞❛❞❡ s❡ ♦ ❥♦❣❛❞♦r ❆ s❡♠♣r❡ t❡♠❡str❛té❣✐❛ ✈❡♥❝❡❞♦r❛✳ ▼✉✐t❛s ✈❡③❡s ✐ss♦ ♥♦s ♣❡r♠✐t❡ ❞❡s❝r❡✈❡r ❡ss❛ ♣r♦♣r✐❡❞❛❞❡s❡♠ ✈ár✐♦s ❣r❛✉s ❞❡ ✧❢♦rç❛✧❞✐st✐♥t♦s✱ ♦ q✉❡ ♣♦ss✐✈❡❧♠❡♥t❡ s❡r✐❛ ❝♦♠♣❧✐❝❛❞♦ ❞❡ s❡❢❛③❡r s❡♠ r❡❝♦rr❡r ❛♦ ❥♦❣♦✳ ❊ss❡ ♥♦♠❡ ❢♦✐ ✐♥tr♦❞✉③✐❞♦ ❡♠ ✶✾✺✻✲✼ ♣♦r ❈✳ ❇❡r❣❡ ❡❆✳ ❘✳ P❡❛rs ❬✾❪✳ ➱ ✉♠❛ té❝♥✐❝❛ ❧❛r❣❛♠❡♥t❡ ✉s❛❞❛✱ t❛♥t♦ ♣❛r❛ ♣r♦✈❛r t❡♦r❡♠❛s✱q✉❛♥t♦ ♣❛r❛ ❝❛r❛❝t❡r✐③❛r ♣r♦♣r✐❡❞❛❞❡s t♦♣♦❧ó❣✐❝❛s ✭✈❡r ❬✶✵❪✮✳

❈r♦♥♦❧♦❣✐❝❛♠❡♥t❡✱ t❡♠♦s ✉♠❛ ❧✐st❛ ❞❡ ❥♦❣♦s t♦♣♦❧ó❣✐❝♦s ❞❡✜♥✐❞♦s ❡♥tr❡ ❞♦✐s❥♦❣❛❞♦r❡s❀ ❝♦♠♦ ❥♦❣♦ ❞❡ ❙✐❡r♣✐♥s❦② ✶✾✷✹✱ ❥♦❣♦ ❞❡ ❇❛♥❛❝❤✲▼❛♥③✉r ✶✾✸✺✱ ❥♦❣♦ ❞❡❯❧❛♠ ✶✾✸✺ ✭é ✉♠❛ ♠♦❞✐✜❝❛çã♦ ❞♦ ❥♦❣♦ ❞❡ ❇❛♥❛❝❤✲▼❛♥③✉r✮✱ ❥♦❣♦ ❞❡ ❇❛♥❛❝❤ ✶✾✸✺✱❥♦❣♦ ❞❡ ❈❤♦q✉❡t ✶✾✸✼✱ ❥♦❣♦ ♣♦♥t♦ ❛❜❡rt♦ ✶✾✼✵ ✭❡st❡ ❥♦❣♦ ♥♦s ❝❤❛♠❛r❡♠♦s ❞❡ ❥♦❣♦●r✉❡♥❤❛❣❡✱ ❡ s❡rá ❞❡ ✐♥t❡r❡ss❡ ♥❡st❡ tr❛❜❛❧❤♦✮✳ ❚♦❞♦s ❡ss❡s ❥♦❣♦s sã♦ ❞❡✜♥✐❞♦s♥✉♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦✳ P❛r❛ ♠❛✐s ❞❡t❛❧❤❡ ❡ ❛ ❝♦♥tr✐❜✉✐çã♦ ❞❡st❡s ❥♦❣♦s ✭✈❡r❬✾✱ ✶✵❪✮✳

❖s ❥♦❣♦s t♦♣♦❧ó❣✐❝♦ ❝✐t❛❞♦s ❛❝✐♠❛ ❢♦r❛♠ ❞❡✜♥✐❞♦s ♥✉♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦✳❊st❡ tr❛❜❛❧❤♦✱ ❛♣r❡s❡♥t❛ ✉♠ ❥♦❣♦ t♦♣♦❧ó❣✐❝♦ ❞❡✜♥✐❞♦ ♥✉♠ ❡s♣❛ç♦ ✉♥✐❢♦r♠❡✱ ♦q✉❛❧ ❢♦✐ ♣✉❜❧✐❝❛❞♦ r❡❝❡♥t❡♠❡♥t❡ ♥♦ ✷✵✶✹ ♣♦r ❏✳ ❇❡❧❧ ❡♠ ❬✷❪✱ ❡ ❡st❛rá ❝♦♠♣♦st♦ ❞❡❝✐♥❝♦ ❝❛♣ít✉❧♦s q✉❡ sã♦ ❞❡s❝r✐t❛s ❡♠ s❡❣✉✐❞❛✳

❖ ♣r✐♠❡✐r♦ ❝❛♣ít✉❧♦ ❝♦♥st❛ ❞❡ ✉♠❛ ❜r❡✈❡ r❡✈✐sã♦ ❞❡ ❡s♣❛ç♦s t♦♣♦❧ó❣✐❝♦s✱❜❛s❡ t♦♣♦❧ó❣✐❝❛s✱ ❣❡r❛çã♦ ❞❡ t♦♣♦❧♦❣✐❛s ❛tr❛✈és ❞❡ ♣r♦♣r✐❡❞❛❞❡s ❞❛❞❛s✱ t♦♣♦❧♦❣✐❛✐♥✐❝✐❛❧✱ t♦♣♦❧♦❣✐❛ ♣r♦❞✉t♦ ❡ ❡s♣❛ç♦s t♦♣♦❧ó❣✐❝♦s ❞❡✜♥✐❞♦s ❡♠ ❢✉♥çã♦ ❞❡ ❝♦❜❡rt✉r❛s✳❆s ♣r✐♥❝✐♣❛✐s r❡❢❡rê♥❝✐❛s sã♦ ❬✻✱ ✹✱ ✽✱ ✼✱ ✸❪✳

❖ s❡❣✉♥❞♦ ❝❛♣ít✉❧♦✱ ❛♣r❡s❡♥t❛ ✉♠❛ ❜r❡✈❡ ✐♥tr♦❞✉çã♦ ❞❡ ❡s♣❛ç♦ ✉♥✐❢♦r♠❡s✱♠✉♥✐❞♦ ❞❡ ❡①❡♠♣❧♦s ♣❛r❛ s✉❛ ♠❡❧❤♦r ❝♦♠♣r❡❡♥sã♦✱ ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s ❜ás✐❝❛s❡ r❡❧❛ç♦❡s ❡♥tr❡ t♦♣♦❧♦❣✐❛ ❡ ✉♥✐❢♦r♠✐❞❛❞❡✳ ❆s r❡❢❡rê♥❝✐❛s ♣r✐♥❝✐♣❛✐s ❬✻✱ ✹✱ ✼✱ ✸❪✳

◆♦ t❡r❝❡✐r♦ ❝❛♣ít✉❧♦✱ s❡rá ❞❡✜♥✐❞♦ ♦ ♣r✐♥❝✐♣❛❧ ❥♦❣♦ ❞❡st❡ tr❛❜❛❧❤♦ ❝❤❛♠❛❞♦❞❡ ❥♦❣♦ ♣r♦①✐♠❛❧ ❞❡✜♥✐❞♦ ♥✉♠ ❡s♣❛ç♦ ✉♥✐❢♦r♠❡✳ ❊st❡ ❥♦❣♦ é r❡❛❧✐③❛❞♦ ❡♥tr❡ ❞♦✐s❥♦❣❛❞♦r❡s A ❡ B ♥✉♠ ❡s♣❛ç♦ ✉♥✐❢♦r♠❡ X✱ ♦♥❞❡ ❛s ❡s❝♦❧❤❛s ❞♦ ❥♦❣❛❞♦r A sã♦❡❧❡♠❡♥t♦s ❞❛ ✉♥✐❢♦r♠✐❞❛❞❡ ❡ ❛s ❡s❝♦❧❤❛s ❞♦ ❥♦❣❛❞♦r B sã♦ ♣♦♥t♦s ❞❡ X✳ ❆❧é♠❞✐ss♦ s❡rá ❞❡✜♥✐❞♦ ❛❧❣✉♥s ❡❧❡♠❡♥t♦s ❞♦ ❥♦❣♦ ♣r♦①✐♠❛❧ ❡ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱ ♦

✶

✷

❡s♣❛ç♦ ♣r♦①✐♠❛❧ ❡♠ ❢✉♥çã♦ ❞♦ ❥♦❣♦ q✉❡ ♣❡r♠✐t❡ ❝❛r❛❝t❡r✐③❛r ❡s♣❛ç♦s ♠étr✐❝♦s❝♦♠♣❧❡t♦✳ ❊st❡ ❝❛♣ít✉❧♦ ❡stá ❜❛s❡❛❞♦ ❡♠ ❬✷❪✳

❖ q✉❛rt♦ ❝❛♣ít✉❧♦✱ ❛♣r❡s❡♥t❛ ♦ ❥♦❣♦ ●r✉❡♥❤❛❣❡ ♦✉ ♣♦♥t♦ ❛❜❡rt♦✱ ❞❡✜♥✐❞♦ ♥✉♠❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ X✱ ♦♥❞❡ ❛s ❡s❝♦❧❤❛s ❞♦ ❥♦❣❛❞♦r A sã♦ ❛❜❡rt♦s ❞❛ t♦♣♦❧♦❣✐❛ ❡ ❛s❡s❝♦❧❤❛s ❞♦ ❥♦❣❛❞♦r B sã♦ ♣♦♥t♦s ❞❡ X✳ ❆❧❡é♠ ❞✐ss♦ s❡rá ❡❧❡♠❡♥t♦s ❞❡st❡ ❥♦❣♦ ❡♦ W ✲❡s♣❛ç♦ ❡♠ ❢✉♥çã♦ ❞❡ ❥♦❣♦ ●r✉❡♥❤❛❣❡✳ ➪ r❡❢❡rê♥❝✐❛ ♣❛r❛ ❡st❡ ❝❛♣ít✉❧♦ é ❬✺❪✳

❋✐♥❛❧♠❡♥t❡✱ ♦ ✉❧t✐♠♦ ❝❛♣ít✉❧♦ ❛♣r❡s❡♥t❛ ✉♠ ❡♥❢r❛q✉❡❝✐♠❡♥t♦ ❞♦ ❥♦❣♦ ♣r♦①✐♠❛❧❡ ❛s ❝♦♥s❡q✉ê♥❝✐❛s t♦♣♦❧ó❣✐❝❛s ❞❡st❡ ❥♦❣♦ ❝♦♠♦ ✐♠♣❧✐❝❛çõ❡s ❛ ❡s♣❛ç♦s t♦♣♦❧ó❣✐❝♦s❝♦❧❡çã♦ ♥♦r♠❛❧✱ ❡s♣❛ç♦s t♦♣♦❧ó❣✐❝♦s ♠❡t❛✲❝♦♠♣❛❝t♦s ❡ ❝♦❧❡çã♦ ❍❛✉s❞♦r✛✳ ❆r❡❢❡rê♥❝✐❛ ♣❛r❛ ❡st❡ ❝❛♣ít✉❧♦ t❛♠❜é♠ é ❬✷❪✳

✷

❈❛♣ít✉❧♦ ✶

Pr❡❧✐♠✐♥❛r❡s

◆❡st❡ ❝❛♣ít✉❧♦✱ ❞❛r❡♠♦s ✉♠❛ ❜r❡✈❡ r❡✈✐sã♦ s♦❜r❡ ❛❧❣✉♥s ❝♦♥❝❡✐t♦s ❜ás✐❝♦s❞❡ t♦♣♦❧♦❣✐❛✱ ♥❡❝❡ssár✐♦s ♣❛r❛ ❝♦♠♣r❡❡♥❞❡r ♦ t❡♠❛ ❞✐ss❡rt❛❞♦✳ ❆s ♣r✐♥❝✐♣❛✐sr❡❢❡rê♥❝✐❛s sã♦ ❬✻✱ ✹✱ ✾✱ ✽✱ ✼✱ ✸❪✳

✶✳✶ ❚♦♣♦❧♦❣✐❛ ❡ ❇❛s❡

❖ ❝♦♥❝❡✐t♦ ❞❡ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ ❢♦✐ ✐♥tr♦❞✉③✐❞♦ ♣♦r ❋❡❧✐① ❍❛✉s❞♦r✛ ❡♠ ✶✾✶✹ ♦q✉❛❧ ❤♦❥❡ é ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ❡s♣❛ç♦ ❞❡ ❍❛✉s❞♦r✛✳ ❆ t♦♣♦❧♦❣✐❛ ♠♦❞❡r♥❛ é ❜❛s❡❛❞❛♥❛ t❡♦r✐❛ ❞♦s ❝♦♥❥✉♥t♦s✱ ❛ss✐♠ ❝♦♠♦ ❛ ♠❛✐♦r ♣❛rt❡ ❞❛ ▼❛t❡♠át✐❝❛✳ ❖ ❝♦♥❝❡✐t♦❞❡ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ é ♠❛✐s ❣❡r❛❧ ❞♦ q✉❡ ♦ ❝♦♥❝❡✐t♦ ❞❡ ❡s♣❛ç♦ ❞❡ ❍❛✉s❞♦r✛ ❡ ❢♦✐✐♥tr♦❞✉③✐❞♦ ♣♦r ❑❛③✐♠✐❡r③ ❑✉r❛t♦✇s❦✐ ❡♠ ✶✾✷✷✳

❉❡✜♥✐çã♦ ✶✳✶✳ ❙❡❥❛ X ✉♠ ❝♦♥❥✉♥t♦ ♥ã♦ ✈❛③✐♦✳ ❯♠❛ ❢❛♠í❧✐❛ T ❞❡ s✉❜❝♦♥❥✉♥t♦s❞❡ X é ✉♠❛ t♦♣♦❧♦❣✐❛ ❡♠ X✱ s❡ t❡♠ ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿

✶✮ ∅✱ X ♣❡rt❡♥❝❡♠ ❛ T❀

✷✮ ❙❡ A1, A2 ∈ T✱ ❡♥tã♦ A1 ∩ A2 ∈ T❀

✸✮ ❙❡ A ⊆ T✱ ❡♥tã♦⋃

A∈A

A ∈ T✳

❖ ♣❛r (X,T) é ❝❤❛♠❛❞♦ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦✱ ♦♥❞❡ T é ✉♠❛ t♦♣♦❧♦❣✐❛ ❡♠ X ❡ ♦s❡❧❡♠❡♥t♦s ❞❡ T sã♦ ❝❤❛♠❛❞♦s ❝♦♥❥✉♥t♦s ❛❜❡rt♦s✳

❉❛q✉✐ ❡♠ ❛❞✐❛♥t❡✱ ❛ ♥♦t❛çã♦ ❞❡ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ s❡rá X✱ ❛ss✉♠✐r❡♠♦sq✉❡ ✐♠♣❧✐❝✐t❛♠❡♥t❡ ❡stá ♠✉♥✐❞♦ ❞❡ ❛❧❣✉♠❛ t♦♣♦❧♦❣✐❛✳ ❊♠ ♦✉tr❛s ♦❝❛s✐õ❡s s❡rá❞❡♥♦t❛❞❛ ♣♦r (X,T)✱ q✉❛♥❞♦ s❡ ♣r❡❝✐s❡ ❡♠ ❛❧❣✉♠❛s ❞❡✜♥✐çõ❡s ♣❛r❛ s❡✉ ♠❡❧❤♦r❡♥t❡♥❞✐♠❡♥t♦✳

❉❡✜♥✐çã♦ ✶✳✷✳ ❙❡❥❛ (X,T) ✉♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦✳ ❯♠❛ s✉❜❢❛♠í❧✐❛ B ❞❡ T é ✉♠❛❜❛s❡ ♣❛r❛ T✱ s❡ ♣❛r❛ t♦❞♦ A ∈ T✱ ❡①✐st❡ B

′

⊂ B✱ t❛❧ q✉❡ A =⋃

B∈B′

B✱ ♦♥❞❡ ♦s

❡❧❡♠❡♥t♦s ❞❡ B sã♦ ❝❤❛♠❛❞♦s ❜ás✐❝♦s✳

✸

✹ ✶✳✶✳ ❚❖P❖▲❖●■❆ ❊ ❇❆❙❊

❉❡✜♥✐çã♦ ✶✳✸✳ ❙❡❥❛ (X,T) ✉♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦✳ ❯♠❛ s✉❜❢❛♠í❧✐❛ S ❞❡ T é ✉♠❛s✉❜✲❜❛s❡ ♣❛r❛ T s❡ B = {

⋂

S∈S′

S : S′

⊆ S} é ✉♠❛ ❜❛s❡ ♣❛r❛ T✳

❖❜s❡r✈❛çã♦ ✶✳✹✳ ❆ ❉❡✜♥✐çã♦ ✶✳✸ q✉❡r ❞✐③❡r q✉❡ S é ✉♠❛ s✉❜✲❜❛s❡ ♣❛r❛ ❛t♦♣♦❧♦❣✐❛ T s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ♣❛r❛ t♦❞♦ A ∈ T − {∅} ❡ t♦❞♦ x ∈ A ❡①✐st❡ S

′

✱s✉❜❝♦♥❥✉♥t♦ ♥ã♦ ✈❛③✐♦ ❞❡ S t❛❧ q✉❡ x ∈

⋂

S∈S′

S ⊆ A✳

❆ ❝♦♥t✐♥✉❛çã♦✱ ❡♥✉♥❝✐❛r❡♠♦s ✉♠❛ ♣r♦♣♦s✐çã♦ q✉❡ s❡r✈❡ ♣❛r❛ ❣❡r❛r t♦♣♦❧♦❣✐❛♥✉♠ ❝♦♥❥✉♥t♦ q✉❛❧q✉❡r✳

❆s ✐❞❡✐❛s ❡①♣r❡ss❛❞❛s ❡ ❛♥❛❧✐s❛❞❛s s♦❜r❡ t♦♣♦❧♦❣✐❛✱ ❜❛s❡ ❡ s✉❜✲❜❛s❡ sã♦✉♠❛ ♠♦t✐✈❛çã♦ ♣❛r❛ ♦❜t❡r ♠ét♦❞♦s ♣❛r❛ ❝♦♥str✉✐r t♦♣♦❧♦❣✐❛s ♥✉♠ ❝♦♥❥✉♥t♦q✉❛❧q✉❡r ❛ ♣❛rt✐r ❞❡ s✉❜✲❝♦❧❡çõ❡s ❞♦ s❡✉ ❝♦♥❥✉♥t♦ ❞❡ ♣❛rt❡s q✉❡ s❛t✐s❢❛③❡♠ ❝❡rt❛s♣r♦♣r✐❡❞❛❞❡s✳

❉❡✜♥✐çã♦ ✶✳✺✳ ❙❡❥❛ X ✉♠ ❝♦♥❥✉♥t♦ ♥ã♦ ✈❛③✐♦✳ ❯♠❛ ❢❛♠í❧✐❛ B ❞❡ s✉❜❝♦♥❥✉♥t♦s❞❡ X é ✉♠❛ ❜❛s❡ t♦♣♦❧ó❣✐❝❛ ❡♠ X✱ s❡ t❡♠ ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿

✶✮ X =⋃

B∈B

B❀

✷✮ ❙❡ B1, B2 ∈ B ❡ x ∈ B1∩B2✱ ❡♥tã♦ ❡①✐st❡ B3 ∈ B t❛❧ q✉❡ x ∈ B3 ⊆ B1∩B2✱

♦♥❞❡ ♦s ❡❧❡♠❡♥t♦s ❞❡ B sã♦ ❝❤❛♠❛❞♦s ❜ás✐❝♦s✳

❊st❡ r❡s✉❧t❛❞♦ ♥♦s ♣r♦♣♦r❝✐♦♥❛ ✉♠ ♠ét♦❞♦ ♣❛r❛ ❣❡r❛r ✉♠❛ t♦♣♦❧♦❣✐❛ ♥✉♠❝♦♥❥✉♥t♦ ✉t✐❧✐③❛♥❞♦ ✉♠❛ ❝♦❧❡çã♦ ❞❡ ❝♦♥❥✉♥t♦s✱ ❞❡ t❛❧ ❢♦r♠❛ q✉❡ ❡❧❛ r❡s✉❧t❡ s❡r✉♠❛ ❜❛s❡ ❞❛ t♦♣♦❧♦❣✐❛ ❣❡r❛❞❛✳

Pr♦♣♦s✐çã♦ ✶✳✻✳ ❙❡❥❛ X ✉♠ ❝♦♥❥✉♥t♦ ♥ã♦ ✈❛③✐♦ ❡ B ✉♠❛ ❢❛♠í❧✐❛ ♥ã♦ ✈❛③✐❛ ❞❡s✉❜❝♦♥❥✉♥t♦s ❞❡ X✳

✶✮ ❙❡ B é ✉♠❛ ❜❛s❡ t♦♣♦❧ó❣✐❝❛ ❡♠ X✱ ❡♥tã♦ TB = {A ⊂ X : ∃B′

⊂ B, A =⋃

B∈B′

B} é ✉♠❛ t♦♣♦❧♦❣✐❛ ❡♠ X✳

✷✮ ❙❡ T é ✉♠❛ t♦♣♦❧♦❣✐❛ ❡♠ X✱ B ⊆ T ❡ T = TB✱ ❡♥tã♦ B é ✉♠❛ ❜❛s❡ t♦♣♦❧ó❣✐❝❛❡♠ X✳

❉❡♠♦♥str❛çã♦✳ ❱❡r Pr♦♣♦s✐çã♦ ✶✳✸✻✱ ♣❣ ✸✸ ❞❡ ❬✽❪

❖ s❡❣✉✐♥t❡ ❝♦♥❝❡✐t♦ é ❞❡ ♠✉✐t❛ ✉t✐❧✐❞❛❞❡ ♣❛r❛ ❛ ❛♥á❧✐s❡ ❞❛s ♣r♦♣r✐❡❞❛❞❡st♦♣♦❧ó❣✐❝❛s ❧♦❝❛✐s ♥✉♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦✳ ■♥t✉✐t✐✈❛♠❡♥t❡✱ ♦ s✐st❡♠❛ ❞❡✈✐③✐♥❤❛♥ç❛s ❞❡ ✉♠ ♣♦♥t♦ ❞❡t❡r♠✐♥❛ ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❛ t♦♣♦❧♦❣✐❛ ♣❡rt♦ ❞♦♣♦♥t♦✳

❉❡✜♥✐çã♦ ✶✳✼✳ ❙❡❥❛ (X,T) ✉♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ ❡ x ✉♠ ❡❧❡♠❡♥t♦ ❞❡ X✳ ❯♠s✉❜❝♦♥❥✉♥t♦ V ❞❡ X é ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❞❡ x ❡♠ (X,T)✱ s❡ ❡①✐st❡ A ∈ T t❛❧ q✉❡x ∈ A ⊆ V ✳

✹

✺ ✶✳✶✳ ❚❖P❖▲❖●■❆ ❊ ❇❆❙❊

◆♦t❛çã♦ ✶✳✽✳ ❖ ❝♦♥❥✉♥t♦ ❞❡ t♦❞❛s ❛s ✈✐③✐♥❤❛♥ç❛s ❞❡ x é ❝❤❛♠❛❞❛ s✐st❡♠❛s ❞❡

✈✐③✐♥❤❛♥ç❛s ❞❡ x ❡ ❞❡♥♦t❡✲s❡ ♣♦r V(x)✳

❉❡✜♥✐çã♦ ✶✳✾✳ ❙❡❥❛♠ (X,T) ✉♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦✱ x ✉♠ ❡❧❡♠❡♥t♦ ❞❡ X ❡V(x) s✐st❡♠❛s ❞❡ ✈✐③✐♥❤❛♥ç❛s ❞❡ x✳ ❯♠❛ ❢❛♠í❧✐❛ B(x) ❞❡ V(x) é ✉♠❛ ❜❛s❡ ❞❡

✈✐③✐♥❤❛♥ç❛s ❞❡ x ❡♠ (X,T)✱ s❡ ♣❛r❛ t♦❞♦ V ∈ V(x) ❡①✐st❡ B ∈ B(x) t❛❧ q✉❡B ⊆ V ✳

❊♠ s❡❣✉✐❞❛✱ ❞❡✜♥✐♠♦s ❜❛s❡ ❧♦❝❛❧ ❞❡ ✈✐③✐♥❤❛♥ç❛s ♦✉ s✐♠♣❧❡s♠❡♥t❡ ❜❛s❡ ❞❡✈✐③✐♥❤❛♥ç❛s✳

❉❡✜♥✐çã♦ ✶✳✶✵✳ ❙❡❥❛ X ✉♠ ❝♦♥❥✉♥t♦ ♥ã♦ ✈❛③✐♦ ❡ x ✉♠ ❡❧❡♠❡♥t♦ ❞❡ X✳ ❯♠❛❢❛♠í❧✐❛ B(x) ❞❡ s✉❜❝♦♥❥✉♥t♦s ❞❡ X✱ ❝♦♠ x ∈ V ∈ B✱ é ✉♠❛ ❜❛s❡ ❞❡ ✈✐③✐♥❤❛♥ç❛s

❞❡ x✱ s❡ t❡♠ ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿

✶✮ ❙❡ V1, V2 ∈ B(x)✱ ❡♥tã♦ ❡①✐st❡ V ∈ B(x) t❛❧ q✉❡✱ V ⊆ V1 ∩ V2✳

✷✮ ❙❡ V ∈ B(x)✱ ❡♥tã♦ ❡①✐st❡ V0 ∈ B(x) t❛❧ q✉❡ s❡ y ∈ V0 ❡♥tã♦ ❡①✐st❡ W ∈ B(y)t❛❧ q✉❡ W ⊆ V ✳

❈♦♠♦ ❝♦♥s❡q✉ê♥❝✐❛ ❞❛ ❉❡✜♥✐çã♦ ✶✳✶✵✱ t❡♠♦s ❛ ❞❡✜♥✐çã♦ ❞❡ ❡s♣❛ç♦s t♦♣♦❧ó❣✐❝♦s♣r✐♠❡✐r♦ ❡♥✉♠❡rá✈❡❧✳

❉❡✜♥✐çã♦ ✶✳✶✶✳ ❯♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ X é ♣r✐♠❡✐r♦ ❡♥✉♠❡rá✈❡❧✱ s❡ ♣❛r❛ t♦❞♦x ∈ X ❡①✐st❡ ✉♠❛ ❜❛s❡ ❧♦❝❛❧ ❞❡ ✈✐③✐♥❤❛♥ç❛s ❞❡ x ❡♠ X✳

❯♠ ❡①❡♠♣❧♦ ❝❧❛r♦ ❞❡ ❡s♣❛ç♦ ♣r✐♠❡✐r♦ ❡♥✉♠❡rá✈❡❧ é✿

❊①❡♠♣❧♦ ✶✳✶✷✳ ❙❡❥❛ R ❡ x ✉♠ ❡❧❡♠❡♥t♦ ❞❡ R✱ ♥♦t❡ q✉❡ ❛ ❢❛♠í❧✐❛ S(x) ❢♦r♠❛❞❛♣❡❧♦s ❡❧❡♠❡♥t♦s ❞❡ ❢♦r♠❛ [x, x+ ε) é ✉♠❛ ❜❛s❡ ❞❡ ✈✐③✐♥❤❛♥ç❛s ❞❡ x ❡♠ R ♣❛r❛ ❛t♦♣♦❧♦❣✐❛ ❞❡ ❙♦r❣❡♥❢r❡②✳ P❛r❛ ♠❛✐s ❞❡t❛❧❤❡ ♣♦❞❡♠♦s ✈❡r ❬✽❪✳

❈♦♥❤❡❝❡♥❞♦ ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❡ ✉♠ s✐st❡♠❛ ❞❡ ✈✐③✐♥❤❛♥ç❛s ❞❡ ✉♠ ♣♦♥t♦✱♣♦❞❡♠♦s s❛❜❡r ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❛ t♦♣♦❧♦❣✐❛✳ ❆ s❡❣✉✐♥t❡ ♣r♦♣♦s✐çã♦ ♥♦s ♦❢❡r❡❝❡✉♠❛ ❢♦r♠❛ ❞❡ ❝♦♥str✉✐r ❡s♣❛ç♦s t♦♣♦❧ó❣✐❝♦s ❛tr❛✈és ❞❡ ❜❛s❡ ❞❡ ✈✐③✐♥❤❛♥ç❛s✳

Pr♦♣♦s✐çã♦ ✶✳✶✸✳ ❙❡❥❛ X ✉♠ ❝♦♥❥✉♥t♦ ♥ã♦ ✈❛③✐♦ t❛❧ q✉❡ ♣❛r❛ ❝❛❞❛ x ❡❧❡♠❡♥t♦❞❡ X t❡♠✲s❡ B(x) ✉♠❛ ❢❛♠í❧✐❛ ❞❡ s✉❜❝♦♥❥✉♥t♦ ❞❡ X✳

✶✮ ❙❡ B(x) é ✉♠❛ ❜❛s❡ ❞❡ ✈✐③✐♥❤❛♥ç❛s ❞❡ x✱ ❡♥tã♦ TB = {A ⊆ X : ∀x ∈A, ∃V ∈ B(x);V ⊆ A} é ✉♠❛ t♦♣♦❧♦❣✐❛ ❡♠ X✳

✷✮ ❙❡ T é ✉♠❛ t♦♣♦❧♦❣✐❛ ❡♠ X ❡ T = TB✱ ❡♥tã♦ B(x) é ✉♠❛ ❜❛s❡ ❞❡ ✈✐③✐♥❤❛♥ç❛s❞❡ x✳

❉❡♠♦♥str❛çã♦✳ ❱❡r Pr♦♣♦s✐çã♦ ✶✳✹✵✱ ♣❣ ✸✽ ❞❡ ❬✽❪✳

✺

✻ ✶✳✷✳ ❚❖P❖▲❖●■❆ ■◆■❈■❆▲ ❊ P❘❖❉❯❚❖

✶✳✷ ❚♦♣♦❧♦❣✐❛ ■♥✐❝✐❛❧ ❡ Pr♦❞✉t♦

◆❡st❛ s❡çã♦✱ ❝♦♥str✉✐r❡♠♦s ✉♠❛ t♦♣♦❧♦❣✐❛ ♥✉♠ ❝♦♥❥✉♥t♦ ❛ ♣❛rt✐r ❞❡ ❢✉♥çõ❡sq✉❛♥❞♦ s❡ t♦r♥❛♠ ❝♦♥tí♥✉❛s✱ ♦ q✉❛❧ ❣❡r❛ ✉♠❛ t♦♣♦❧♦❣✐❛ ♠❡♥♦r✱ ❝✉❥❛s r❡❢❡rê♥❝✐❛sã♦ ❬✻✱ ✹✱ ✽✱ ✼✱ ✸❪✳

❙❡❥❛ Y ✉♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ ❡ X ✉♠ ❝♦♥❥✉♥t♦ ♥ã♦ ✈❛③✐♦✱ ♦ ♣r♦♣ós✐t♦ é♦❜t❡r ✉♠❛ t♦♣♦❧♦❣✐❛ ❡♠ X ✭t♦♣♦❧♦❣✐❛ ♠❛✐s ♣❡q✉❡♥❛ ❡♠ X✮✳ ❈♦♥s✐❞❡r❡ ❛ ❢✉♥çã♦f : X −→ Y t❛❧ q✉❡✱ s❡❥❛ ♣♦ssí✈❡❧ q✉❡ ❛ ❢✉♥çã♦ f s❡ t♦r♥❡ ❝♦♥t✐♥✉❛✳

◆♦t❡ q✉❡ ♣❛r❛ t♦❞❛ ❢✉♥çã♦ f ❞❡✜♥✐❞❛ ♥✉♠ ❝♦♥❥✉♥t♦ X ❡ ❝♦♠ ✈❛❧♦r❡s ❡♠ Y ❡♣❛r❛ t♦❞❛ ❢❛♠í❧✐❛ {Aα : α ∈ I} ❞❡ s✉❜❝♦♥❥✉♥t♦s ❞❡ Y s❡ ❝✉♠♣r❡✿

✶✮ f−1[∅] = ∅ ❡ f−1[Y ] = X

✷✮ f−1[⋃

α∈I

Aα] =⋃

α∈I

f−1[Aα]

✸✮ f−1[⋂

α∈I

Aα] =⋂

α∈I

f−1[Aα]

❆s três ✐❣✉❛❧❞❛❞❡s ❛❝✐♠❛ ♥♦s ♣❡r♠✐t❡♠ ❞❡✜♥✐r ✉♠❛ t♦♣♦❧♦❣✐❛ ❡♠ X✱ q✉❛♥❞♦t❡♠♦s ✉♠❛ ❢✉♥çã♦ f ❞❡✜♥✐❞❛ ❡♠ X ❡ ❝♦♠ ❝♦♥❥✉♥t♦ ❞❡ ✈❛❧♦r❡s ♥✉♠ ❡s♣❛ç♦t♦♣♦❧ó❣✐❝♦ Y ✳

Pr♦♣♦s✐çã♦ ✶✳✶✹✳ ❙❡ (Y,T) é ✉♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦✱ ❡♥tã♦ ♦ ❝♦♥❥✉♥t♦ Tf ={f−1[A] : A ∈ T} é ✉♠❛ t♦♣♦❧♦❣✐❛ ❡♠ X✳

❉❡♠♦♥str❛çã♦✳ ❱❡r Pr♦♣♦s✐çã♦ ✹✳✶✱ ♣❣ ✶✶✸ ❞❡ ❬✽❪✳

◆♦t❛çã♦ ✶✳✶✺✳ ❆ t♦♣♦❧♦❣✐❛ Tf é ❝❤❛♠❛❞❛ ❞❡ t♦♣♦❧♦❣✐❛ ✐♥✐❝✐❛❧ ❡♠ X ❞❡✜♥✐❞❛♣♦r f ❡ Y ♦✉ t♦♣♦❧♦❣✐❛ ❢r❛❝❛ ✐♥❞✉③✐❞❛ ♣♦r f ✳

❖❜s❡r✈❛çã♦ ✶✳✶✻✳ ❆ t♦♣♦❧♦❣✐❛ Tf é ❛ ♠❡♥♦r t♦♣♦❧♦❣✐❛ ✭♦✉ ♠❛✐s ❢r❛❝❛✮ ❞❛st♦♣♦❧♦❣✐❛s ❡♠ X q✉❡ t♦r♥❛♠ ❝♦♥t✐♥✉❛ à ❢✉♥çã♦ f ✳

❆❣♦r❛ ✐❧✉str❛r❡♠♦s ❝♦♠ ✉♠ ❡①❡♠♣❧♦ ❛ t♦♣♦❧♦❣✐❛ ✐♥✐❝✐❛❧✳

❊①❡♠♣❧♦ ✶✳✶✼✳ ❙❡❥❛ E ✉♠ s✉❜❝♦♥❥✉♥t♦ ❞❡ ✉♠ ❝♦♥❥✉♥t♦ X✱ ❡ R ❛ r❡t❛ r❡❛❧ ❝♦♠❛ t♦♣♦❧♦❣✐❛ ✉s✉❛❧✳ ❈♦♥s✐❞❡r❡♠♦s ❛ ❢✉♥çã♦ ❝❛r❛❝t❡ríst✐❝❛ ❞❡ E✱ XE : X −→ R✱❞❛❞❛ ♣♦r✿

XE(x) =

{

0 se x ∈ X − E,1 se x ∈ E.

❆ t♦♣♦❧♦❣✐❛ ✐♥❞✉③✐❞❛ ❡♠ X ♣❡❧❛ ❢✉♥çã♦ ❝❛r❛t❡ríst✐❝❛ XE é ♦ ❝♦♥❥✉♥t♦ ❞❡ ✐♠❛❣❡♥s✐♥✈❡rs❛s ♣♦r XE✱ ❞❡ ❛❜❡rt♦s ❡♠ R ❞❛ ❢♦r♠❛ q✉❡✿

TXE= {∅, X,E,X − E}.

P❛r❛ ✉♠ s✉❜❝♦♥❥✉♥t♦ ❛❜❡rt♦ A ❞❡ R✱ t❡♠♦s✿

✻

✼ ✶✳✷✳ ❚❖P❖▲❖●■❆ ■◆■❈■❆▲ ❊ P❘❖❉❯❚❖

✶✮ ❙❡ {0, 1} ∩ A = ∅✱ ❡♥tã♦ (XE)−1[A] = ∅✳

✷✮ ❙❡ {0, 1} ⊆ A✱ ❡♥tã♦ (XE)−1[A] = X✳

✸✮ ❙❡ {0, 1} ∩ A = 0✱ ❡♥tã♦ (XE)−1[A] = X − E✳

✹✮ ❙❡ {0, 1} ∩ A = 1✱ ❡♥tã♦ (XE)−1[A] = E✳

▼❡♥❝✐♦♥❛r❡♠♦s ✉♠ t❡♦r❡♠❛ ♠✉✐t♦ ✐♠♣♦rt❛♥t❡ ♣❛r❛ ❝♦♥str✉✐r ❛ t♦♣♦❧♦❣✐❛✐♥✐❝✐❛❧✳ ◆❛ t♦♣♦❧♦❣✐❛ ✐♥✐❝✐❛❧ ❛ ♣r✐♦r✐ ♥ã♦ ❝♦♥❤❡❝❡♠♦s ♦s ❡❧❡♠❡♥t♦s ❞❛ t♦♣♦❧♦❣✐❛✱♠❛s t❡♥❞♦ ✉♠❛ s✉❜✲❜❛s❡ s❡ ♣♦❞❡ ❣❡r❛r ❛ t♦♣♦❧♦❣✐❛ ❡ s❛❜❡r ❝♦♠♦ sã♦ ♦s ❡❧❡♠❡♥t♦s✳

❚❡♦r❡♠❛ ✶✳✶✽✳ ❙❡❥❛ {(Xα,Tα) : α ∈ I} ✉♠❛ ❢❛♠í❧✐❛ ❞❡ ❡s♣❛ç♦s t♦♣♦❧ó❣✐❝♦s✱ X✉♠ ❝♦♥❥✉♥t♦ ♥ã♦ ✈❛③✐♦ ❡ F = {fα : X −→ Xα : α ∈ I} ✉♠❛ ❢❛♠í❧✐❛ ❞❡ ❢✉♥çõ❡s✳❊♥tã♦ ♦ ❝♦♥❥✉♥t♦ S = {f−1

α [A] : α ∈ I ❡ A ∈ Tα} é ✉♠❛ s✉❜✲❜❛s❡ ♣❛r❛ ❛t♦♣♦❧♦❣✐❛ TF✳

❉❡♠♦♥str❛çã♦✳ ❱❡r ❚❡♦r❡♠❛ ✹✳✺✱ ♣❣ ✶✷✻ ❞❡ ❬✽❪✳

◆♦t❛çã♦ ✶✳✶✾✳ ❉❡♥♦t❡✲s❡ ♣♦r TF à ♠❡♥♦r ❞❛s t♦♣♦❧♦❣✐❛s ❡♠ X q✉❡ t♦r♥❛♠ ❛t♦❞❛ ❢✉♥çã♦ f ∈ F ♥✉♠❛ ❢✉♥çã♦ ❝♦♥t✐♥✉❛✳

❉❡✜♥✐çã♦ ✶✳✷✵✳ ❙❡❥❛ {(Xα,Tα) : α ∈ I} ✉♠❛ ❢❛♠í❧✐❛ ❞❡ ❡s♣❛ç♦s t♦♣♦❧ó❣✐❝♦s✱ X✉♠ ❝♦♥❥✉♥t♦ ♥ã♦ ✈❛③✐♦ ❡ F = {fα : X −→ Xα : α ∈ I} ✉♠❛ ❢❛♠í❧✐❛ ❞❡ ❢✉♥çõ❡s✳❆ t♦♣♦❧♦❣✐❛ TF ❡♠ X é ❝❤❛♠❛❞❛ ❞❡ t♦♣♦❧♦❣✐❛ ❢r❛❝❛ ♦✉ ✐♥✐❝✐❛❧ ✐♥❞✉③✐❞❛ ♣♦r F✳

❈♦♠♦ ❛♣❧✐❝❛çã♦ ❞❡st❡ r❡s✉❧t❛❞♦ s❡rá ✐♥tr♦❞✉③✐❞♦ ✉♠❛ t♦♣♦❧♦❣✐❛ ♥✉♠ ♣r♦❞✉t♦❝❛rt❡s✐❛♥♦s ❞❡ ❞♦✐s ❡❧❡♠❡♥t♦s✳

❖❜s❡r✈❛çã♦ ✶✳✷✶✳ P❛r❛ t♦❞♦ ♣❛r ❞❡ ❝♦♥❥✉♥t♦ X ❡ Y s❡ ♣♦❞❡ ❝♦♥s✐❞❡r❛r ♦❝♦♥❥✉♥t♦ ♣r♦❞✉t♦ X × Y ✳ ❙❡ X ❡ Y sã♦ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦s✱ ❡♥tã♦ é ♥❛t✉r❛❧t❡♥t❛r ❝♦♥str✉✐r ✉♠❛ t♦♣♦❧♦❣✐❛ ❡♠ X × Y ✳ ❯s❛♥❞♦ ❛ té❝♥✐❝❛ ❞❡ t♦♣♦❧♦❣✐❛ ✐♥✐❝✐❛❧✱❝♦♥s✐❞❡r❡ ❛s ♣r♦❥❡çõ❡s✿

ΠX : X × Y −→ X;

ΠY : X × Y −→ Y,

t❛❧ q✉❡✱ ΠX(x, y) = x ❡ ΠY (x, y) = y r❡s♣❡t✐✈❛♠❡♥t❡✱ ♣❛r❛ t♦❞♦ (x, y) ∈ X × Y ✳

❉❡✜♥✐çã♦ ✶✳✷✷✳ ❙❡❥❛♠ X ❡ Y ❡s♣❛ç♦s t♦♣♦❧ó❣✐❝♦s✳ ❆ t♦♣♦❧♦❣✐❛ TP é ❝❤❛♠❛❞❛t♦♣♦❧♦❣✐❛ ♣r♦❞✉t♦ ♦✉ t♦♣♦❧♦❣✐❛ ❞❡ ❚②❝❤♦♥♦✛✱ ❡♠ X × Y ✐♥❞✉③✐❞❛ ♣♦r P ={ΠX ,ΠY }

❆ t♦♣♦❧♦❣✐❛ TP é ❛ ♠❡♥♦r ❞❛s t♦♣♦❧♦❣✐❛s ❡♠ X × Y q✉❡ t♦r♥❛♠ ΠX ❡ ΠY ❡♠❢✉♥çõ❡s ❝♦♥t✐♥✉❛s✳

❊♠ s❡❣✉✐❞❛✱ ✈❡r❡♠♦s ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s ✐♠♣♦rt❛♥t❡s ♥✉♠ ❡s♣❛ç♦t♦♣♦❧ó❣✐❝♦ ♣r♦❞✉t♦✳

❖❜s❡r✈❛çã♦ ✶✳✷✸✳ ❉❛ ❉❡✜♥✐çã♦ ✶✳✷✷ t❡♠✲s❡✿

✼

✽ ✶✳✷✳ ❚❖P❖▲❖●■❆ ■◆■❈■❆▲ ❊ P❘❖❉❯❚❖

✶✮ ❙❡ X ♦✉ Y é ✈❛③✐♦✱ ❡♥tã♦ TP = ∅✳

✷✮ ❙❡ BX é ✉♠❛ ❜❛s❡ ❡♠ TX ❡ BY é ✉♠❛ ❜❛s❡ ❡♠ TY ✱ ❡♥tã♦ ♦ ❝♦♥❥✉♥t♦{Π−1

X (B) ∩ Π−1Y (C) : B ∈ BX ❡ C ∈ BY } é ✉♠❛ ❜❛s❡ ♣❛r❛ TP✳

❆❣♦r❛ ✈❡r❡♠♦s ❛❧❣✉♥s ❡①❡♠♣❧♦s ✭✈❡r ❬✽✱ ✼❪✮ q✉❡ ✈ã♦ t❡r ♠✉✐t❛ ✐♠♣♦rtâ♥❝✐❛ ❛♦❧♦♥❣♦ ❞♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡st❡s tr❛❜❛❧❤♦✳

❊①❡♠♣❧♦ ✶✳✷✹✳ ❯♠❛ ❜❛s❡ ❞❛ t♦♣♦❧♦❣✐❛ ✉s✉❛❧ ♥❛ r❡t❛ r❡❛❧ R é ❛ ❝♦❧❡çã♦ ❞❡✐♥t❡r✈❛❧♦s ❛❜❡rt♦s (a, b)✳ ❙❡❥❛ Πi : R2 −→ R ❛ ♣r♦❥❡çã♦ ❛♦ i✲❡s✐♠♦ ❢❛t♦r ♣❛r❛i = 1, 2✳ ❆ t♦♣♦❧♦❣✐❛ ♣r♦❞✉t♦ ❡♠ R

2 é ❣❡r❛❞❛ ♣❡❧❛ ❝♦❧❡çã♦ ❞❡ t♦❞♦s ♦s ❝♦♥❥✉♥t♦s❞❛ ❢♦r♠❛✿

{(x, y) ∈ R2 : a < x < b ❡ c < y < d}.

❚♦❞♦ ❡❧❡♠❡♥t♦ ❞❛ t♦♣♦❧♦❣✐❛ ♣r♦❞✉t♦ R2 s❡ ❡①♣r❡ss❛ ❞❛ ❢♦r♠❛✿

Π−11 [(a, b)] ∩ Π−1

2 [(c, d)],

♦♥❞❡

Π−11 [(a, b)] = {(x, y) ∈ R

2 : a < x < b},

Π−12 [(c, d)] = {(x, y) ∈ R

2 : c < y < d}.

❊①❡♠♣❧♦ ✶✳✷✺✳ ❯♠❛ ❜❛s❡ ❞❛ t♦♣♦❧♦❣✐❛ ❞❡ ❙♦r❣❡♥❢r❡② ♥❛ r❡t❛ r❡❛❧ R é ❛ ❝♦❧❡çã♦❞❡ ✐♥t❡r✈❛❧♦s ❞❛ ❢♦r♠❛✿

[a, b) = {(x, y) ∈ R2 : a ≤ x < b}.

❙❡❥❛ Πi : R2 −→ R ❛ ♣r♦❥❡çã♦ ❛♦ i✲❡s✐♠♦ ❢❛t♦r ♣❛r❛ i = 1, 2✳ ❙❡ ♣♦❞❡ ❞❡s❝r❡✈❡r ❛

t♦♣♦❧♦❣✐❛ ♣r♦❞✉t♦ ❡♠ R2✳ ❆ ❝♦❧❡çã♦ B ❝♦♠ ❡❧❡♠❡♥t♦s ❞❛ ❢♦r♠❛✿

Π−11 [[a, b)] ∩ Π−1

2 [[c, d)] = {(x, y) ∈ R2 : a ≤ x < b ❡ c ≤ y < d},

è ✉♠❛ ❜❛s❡ ♣❛r❛ ❛ t♦♣♦❧♦❣✐❛ ❡♠ R2✱ ♦♥❞❡✿

Π−11 [(a, b)] = {(x, y) ∈ R

2 : a ≤ x < b},

Π−12 [(c, d)] = {(x, y) ∈ R

2 : c ≤ y < d}.

❋✐♥❛❧♠❡♥t❡✱ ♣❛r❛ ❢❡❝❤❛r ❡st❛ s❡çã♦✱ ✈❛♠♦s ❣❡♥❡r❛❧✐③❛r ❛ t♦♣♦❧♦❣✐❛ ♣r♦❞✉t♦ ❞❡✉♠❛ ❝♦❧❡çã♦ ✜♥✐t❛ ❞❡ ❡s♣❛ç♦s t♦♣♦❧ó❣✐❝♦s✱ ♣❛r❛ ✉♠❛ ❝♦❧❡çã♦ ✐♥✜♥✐t❛ ❞❡ ❡s♣❛ç♦t♦♣♦❧ó❣✐❝♦s✳ ❇❛s✐❝❛♠❡♥t❡✱ ♣❛r❛ ❡st❡ tr❛❜❛❧❤♦ ♣r❡❝✐s❛r❡♠♦s ❞❡ ✉♠❛ ❝♦❧❡çã♦ ✐♥✜♥✐t❛❞❡ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦s✱ ♣❛r❛ ♦ q✉❛❧ ✐♥tr♦❞✉③✐r❡♠♦s ❞❡ ❢♦r♠❛ ❜r❡✈❡ ♦s s❡❣✉✐♥t❡s❝♦♥❝❡✐t♦s✳

❈♦♥s✐❞❡r❡ I ✉♠ ❝♦♥❥✉♥t♦ ❞❡ í♥❞✐❝❡s✱ ❡ ♦ ♣r♦❞✉t♦ ❝❛rt❡s✐❛♥♦ ❞❛ ❢❛♠í❧✐❛ ❞❡❡s♣❛ç♦s t♦♣♦❧ó❣✐❝♦s {(Xα,Tα) : α ∈ I}✳

∏

α∈I

Xα = {f : I →⋃

α∈I

Xα t❛❧ q✉❡ ∀α ∈ I, f(α) ∈ Xα}.

✽

✾ ✶✳✸✳ ❈❖❇❊❘❚❯❘❆❙ ❉❊ ❊❙P❆➬❖❙

❉❡✜♥✐çã♦ ✶✳✷✻✳ ❙❡❥❛∏

α∈I

Xα ♦ ♣r♦❞✉t♦ ❝❛rt❡s✐❛♥♦ ❞❛ ❢❛♠í❧✐❛ {Xα : α ∈ I}✳

❉❡✜♥❛✲s❡ ❛ ♣r♦❥❡çã♦ s♦❜r❡ ♦ α✲és✐♠♦ ❢❛t♦r ❝♦♠♦

Πα :∏

α∈I

Xα −→ Xα,

t❛❧ q✉❡ ♣❛r❛ t♦❞♦ f ∈∏

α∈I

Xα✱ ❡①✐st❡ α ∈ I ❝♦♠ πα(f) = f(α)✳

◆♦t❛çã♦ ✶✳✷✼✳ ❖ ❝♦♥❥✉♥t♦ ❞❡ ♣r♦❥❡çõ❡s {Πα :∏

α∈I

Xα −→ Xα : α ∈ I} s❡rá

❞❡♥♦t❛❞♦ ♣♦r P✳

❉❡✜♥✐çã♦ ✶✳✷✽✳ ❈❤❛♠❡✲s❡ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ ♣r♦❞✉t♦ ♦✉ ❡s♣❛ç♦ ♣r♦❞✉t♦

❚②❝❤♦♥♦✛ ❞❡ ❡s♣❛ç♦s Xα ♥♦ ♣r♦❞✉t♦∏

α∈I

Xα ♦ ♣❛r✿

(∏

α∈I

Xα,TP),

♦♥❞❡ TP é ✐♥❞✉③✐❞❛ ♣❡❧❛ ❢❛♠í❧✐❛ ❞❡ ♣r♦❥❡çõ❡s P✳

❆ t♦♣♦❧♦❣✐❛ TP é ❛ ♠❡♥♦r ❞❛s t♦♣♦❧♦❣✐❛s ❡♠∏

α∈I

Xα q✉❡ t♦r♥❛♠ t♦❞♦ Πα ❡♠

❢✉♥çõ❡s ❝♦♥t✐♥✉❛s✳ ▲♦❣♦ ✈❡r❡♠♦s ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s ✐♠♣♦rt❛♥t❡s ❞❡ ❡s♣❛ç♦st♦♣♦❧ó❣✐❝♦s ♦❜t✐❞♦ ♣❡❧♦ ♣r♦❞✉t♦s ❚②❝❤♦♥♦✛✳

❚❡♦r❡♠❛ ✶✳✷✾✳ ❙❡❥❛ {(Xα,Tα) : α ∈ I} ✉♠❛ ❢❛♠í❧✐❛ ❞❡ ❡s♣❛ç♦s t♦♣♦❧ó❣✐❝♦s✳❊♥tã♦ ♦ ❝♦♥❥✉♥t♦ S = {Π−1

α [B] : α ∈ I ❡ B ∈ Tα} é ✉♠❛ s✉❜✲❜❛s❡ ❞❛ t♦♣♦❧♦❣✐❛TP✱ ♦♥❞❡ P é ❛ ❢❛♠í❧✐❛ ❞❡ ♣r♦❥❡çõ❡s ❞❛s ❝♦♦r❞❡♥❛❞❛s α ∈ I✳

❉❡♠♦♥str❛çã♦✳ ❱❡r ❚❡♦r❡♠❛ ✹✳✶✺✱ ♣❣ ✶✸✷ ❞❡ ❬✽❪✳

❆ ❝♦♥✈❡r❣ê♥❝✐❛ ❞❡ ✉♠❛ s❡q✉ê♥❝✐❛ ♥✉♠ ❡s♣❛ç♦ ♣r♦❞✉t♦ ❡stá ❞❡t❡r♠✐♥❛❞❛ ♣❡❧❛❝♦♥✈❡r❣ê♥❝✐❛ ❞❛s s❡q✉ê♥❝✐❛s q✉❡ ❞❡t❡r♠✐♥❛ ❝❛❞❛ ♣r♦❥❡çã♦✳ ❆ s❡❣✉✐r✱ ❡♥✉♥❝✐❛r❡♠♦s✉♠❛ ♣r♦♣♦s✐çã♦ ♣❛r❛ ♦ ❝❛s♦ q✉❛♥❞♦ ❛ q✉❡stã♦ ❡♥✈♦❧✈❛ ❝♦♥✈❡r❣ê♥❝✐❛ ❡♠ ❡s♣❛ç♦♣r♦❞✉t♦s✳

Pr♦♣♦s✐çã♦ ✶✳✸✵✳ ❯♠❛ s❡q✉ê♥❝✐❛ (x1, x2, ..., xn, ...) ❞❡ ♣♦♥t♦s ❡♠∏

α∈I

Xα

❝♦♥✈❡r❣❡ ❛ ✉♠ ♣♦♥t♦ ❞❡ x ❞❡∏

α∈I

Xα✱ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❛ s❡q✉ê♥❝✐❛

(Πα(x1),Πα(x2), ...,Πα(xn), ...) ❝♦♥✈❡r❣❡ ❛ Πα(x) ❡♠ Xα✱ ♣❛r❛ t♦❞♦ α ∈ I✳

❉❡♠♦♥str❛çã♦✳ ❱❡r Pr♦♣♦s✐çã♦ ✹✳✶✼✱ ♣❣ ✶✸✹ ❞❡ ❬✽❪✳

✶✳✸ ❈♦❜❡rt✉r❛s ❞❡ ❊s♣❛ç♦s

◆❡st❛ s❡❝çã♦✱ ♦ ♦❜❥❡t✐✈♦ é ❛♣r❡s❡♥t❛r ❞❡✜♥✐çõ❡s ❡ ❛❧❣✉♥s ❡①❡♠♣❧♦ ❡ ✐♠♣❧✐❝❛çõ❡s❞❡ ❡s♣❛ç♦s t♦♣♦❧ó❣✐❝♦s✱ q✉❡ ❡stã♦ ❞❡✜♥✐❞♦s ♣♦r ♠❡✐♦ ❞❡ ❝♦❜❡rt✉r❛s✱ ❝♦♠♦

✾

✶✵ ✶✳✸✳ ❈❖❇❊❘❚❯❘❆❙ ❉❊ ❊❙P❆➬❖❙

❡s♣❛ç♦ ❝♦♠♣❛❝t♦s✱ ♣❛r❛✲❝♦♠♣❛❝t♦s✱ ♠❡t❛✲❝♦♠♣❛❝t♦s✱ ❝♦❧❡çã♦ ❍❛✉s❞♦r✛ ❡ ❝♦❧❡çã♦♥♦r♠❛❧✳

❆ s❡❣✉✐♥t❡ ❞❡✜♥✐çã♦ ❞❡ ♣❛r❛✲❝♦♠♣❛❝✐❞❛❞❡ ❢♦r♥❡❝❡ ♣r♦♣r✐❡❞❛❞❡s ❧♦❝❛✐s✳

❉❡✜♥✐çã♦ ✶✳✸✶✳ ❙❡❥❛♠ X ✉♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦✱ U ✉♠❛ ❝♦❜❡rt✉r❛ ❛❜❡rt❛ ❞❡ X✳❯♠❛ ❢❛♠í❧✐❛ V ❞❡ s✉❜❝♦♥❥✉♥t♦s ❞❡ X é ✉♠ r❡✜♥❛♠❡♥t♦ ❛❜❡rt♦ ❧♦❝❛❧♠❡♥t❡

✜♥✐t♦ ❞❛ ❝♦❜❡rt✉r❛ ❛❜❡rt❛ U✱ s❡ s❛t✐s❢❛③ ❛s s❡❣✉✐♥t❡ ♣r♦♣r✐❡❞❛❞❡s✿

✶✮ P❛r❛ t♦❞♦ V ∈ V ❡①✐st❡ U ∈ U t❛❧ q✉❡ V ⊆ U ✳

✷✮ P❛r❛ t♦❞♦ x ∈ X ❡①✐st❡♠ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ V (x) ❞❡ x ❡ V′

⊆ V ✜♥✐t♦✱ t❛❧q✉❡ ♣❛r❛ t♦❞♦ V ∈ V

′

✱ t❡♠✲s❡ q✉❡ V (x) ∩ V 6= ∅✳

❖❜s❡r✈❡ q✉❡ s❡ ❛ ❢❛♠í❧✐❛ V ♥❛ ❉❡✜♥✐çã♦ ✶✳✸✶ só s❛t✐s❢❛③ ❛ ♣r♦♣r✐❡❞❛❞❡ ✶✮✱❡♥tã♦ V é ✉♠ r❡✜♥❛♠❡♥t♦ ❛❜❡rt♦ ❞❡ U✳

❉❡✜♥✐çã♦ ✶✳✸✷✳ ❯♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ X é ♣❛r❛✲❝♦♠♣❛❝t♦✱ s❡ ♣❛r❛ t♦❞❛❝♦❜❡rt✉r❛ ❛❜❡rt❛ U ❞❡ X✱ ❡①✐st❡ ✉♠ r❡✜♥❛♠❡♥t♦ ❛❜❡rt♦ ❧♦❝❛❧♠❡♥t❡ ✜♥✐t♦ V ❞❡U✳

❯♠ ❡①❡♠♣❧♦ ❝❧❛r♦ ❞❡ ❡s♣❛ç♦s ♣❛r❛✲❝♦♠♣❛❝t♦s é ✉♠ ❡s♣❛ç♦ ❝♦♠♣❛❝t♦ ✭✉♠❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ X é ❝♦♠♣❛❝t♦✱ s❡ ♣❛r❛ t♦❞❛ ❝♦❜❡rt✉r❛ ❛❜❡rt❛ ❞❡ X✱ ❡①✐st❡ ✉♠❛s✉❜✲❝♦❜❡rt✉r❛ ✜♥✐t❛ ❞❡ X✮✳ P❛r❛ ♠❛✐♦r❡s ❞❡t❛❧❤❡s ✈❡r ❬✽❪✳

❆s s❡❣✉✐♥t❡s ❞❡✜♥✐çõ❡s ❞❡ ♠❡t❛✲❝♦♠♣❛❝✐❞❛❞❡ ❡ ❝♦❧❡çã♦ ♥♦r♠❛❧ ❡ ❝♦❧❡çã♦❍❛✉❞♦r✛ t❛♠❜é♠ ❡stã♦ ❞❡✜♥✐❞❛s ❡♠ ❢✉♥çã♦ ❞❡ ❝♦❜❡rt✉r❛s✳

❉❡✜♥✐çã♦ ✶✳✸✸✳ ❯♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ X é ✉♠❛ ❝♦❧❡çã♦ ♥♦r♠❛❧✱ s❡ ♣❛r❛ t♦❞❛❢❛♠í❧✐❛ ❢❡❝❤❛❞❛ ❞✐s❥✉♥t❛ ❞♦✐s ❛ ❞♦✐s F ❡①✐st❡ ✉♠❛ ❢❛♠í❧✐❛ ❛❜❡rt❛ ❞✐s❥✉♥t❛ ❞♦✐s ❛❞♦✐s✱ U✱ t❛❧ q✉❡ ♣❛r❛ t♦❞♦ F ∈ F t❡♠✲s❡ F ⊂ A✱ ❝♦♠ A ∈ U✳

❇❛s✐❝❛♠❡♥t❡ ❛ ❉❡✜♥✐çã♦ ✶✳✸✸ ❞❡r✐✈❛ ❞❛ ❞❡✜♥✐çã♦ ❞❡ ✉♠ ❡s♣❛ç♦ ♥♦r♠❛❧ ✭✉♠❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ X é ♥♦r♠❛❧✱ s❡ ♣❛r❛ t♦❞♦ F1 ❡ F2 ❢❡❝❤❛❞♦s ❞✐s❥✉♥t♦s ❡♠ X❡①✐st❡♠ ❛❜❡rt♦s ❞✐s❥✉♥t♦s A1 ❡ A2 t❛❧ q✉❡ F1 ⊆ A1 ❡ F2 ⊆ A2✮✳ P❛r❛ ♠❛✐♦r❡s❞❡t❛❧❤❡s ✈❡r ❬✽❪✳

❉❡✜♥✐çã♦ ✶✳✸✹✳ ❯♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ X é ♠❡t❛✲❝♦♠♣❛❝t♦ ❡♥✉♠❡rá✈❡❧✱ s❡♣❛r❛ t♦❞❛ ❢❛♠í❧✐❛ ❞❡ ❝♦♥❥✉♥t♦s ❢❡❝❤❛❞♦s F ❞❡ X✱ t❛✐s q✉❡✿

F1 ⊇ F2 ⊇, ...,⊇ Fn ⊇, ... ❡⋂

n∈N

Fn = ∅;

❡①✐st❡ ✉♠❛ ❢❛♠í❧✐❛ ❞❡ ❝♦♥❥✉♥t♦s ❛❜❡rt♦s U ❞❡ X✱ t❛✐s q✉❡

U1 ⊇ U2 ⊇, ...,⊇ Un ⊇, ... ❡⋂

n∈N

Un = ∅,

♦♥❞❡ ♣❛r❛ t♦❞♦ n ∈ N✱ t❡♠✲s❡ q✉❡ Fn ⊆ Un✳

✶✵

✶✶ ✶✳✸✳ ❈❖❇❊❘❚❯❘❆❙ ❉❊ ❊❙P❆➬❖❙

❋✐♥❛❧♠❡♥t❡✱ t❡♠♦s ❛ ❞❡✜♥✐çã♦ ❝♦❧❡çã♦ ❍❛✉s❞♦r✛ q✉❡ s❡ ❞❡r✐✈❛ ❞❛ ❞❡✜♥✐çã♦ ❞❡❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ ❍❛✉s❞♦r✛✳

❉❡✜♥✐çã♦ ✶✳✸✺✳ ❯♠ t♦♣♦❧ó❣✐❝♦ X é ❍❛✉s❞♦r✛ s❡ ♣❛r❛ t♦❞♦ x, y ∈ X ❝♦♠ x 6= y✱❡①✐st❡ ✈✐③✐♥❤❛♥ç❛s ❛❜❡rt❛s U ✱ V ❞❡ x ❡ y r❡s♣❡t✐✈❛♠❡♥t❡ t❛❧ q✉❡ U ∩ V = ∅✳

P❛r❛ ♠❛✐♦r❡s ❞❡t❛❧❤❡s ✈❡r ❬✽✱ ✼✱ ✸❪✳

❉❡✜♥✐çã♦ ✶✳✸✻✳ ❯♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ X é ✉♠❛ ❝♦❧❡çã♦ ❍❛✉s❞♦r✛✱ s❡ ♣❛r❛t♦❞♦ ❝♦♥❥✉♥t♦ ❢❡❝❤❛❞♦ ❞✐s❝r❡t♦ F ❡♠ X ❡①✐st❡ ✉♠❛ ❝♦❧❡çã♦ ❛❜❡rt❛ ❞✐s❥✉♥t❛ ❞♦✐s ❛❞♦✐s U t❛❧ q✉❡ ♣❛r❛ t♦❞♦ x ∈ F ❡①✐st❡ U ∈ U✱ ❝♦♠ x ∈ U ✳

❈♦♠♦ ❡①❡♠♣❧♦s ❞❡ ❡s♣❛ç♦s ❝♦❧❡çã♦ ❍❛✉s❞♦r✛ t❡♠♦s ♦s ❡s♣❛ç♦s ❡✉❝❧✐❞✐❛♥♦s ❡❡s♣❛ç♦s ♠étr✐❝♦s ❡♠ ❣❡r❛❧✳ ❆s ❞❡✜♥✐çõ❡s ❞❡st❛ s❡çã♦✱ s❡rã♦ ❞❡ ♠✉✐t❛ ✐♠♣♦rtâ♥❝✐❛♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞♦ ❝❛♣ít✉❧♦ ✜♥❛❧✳

✶✶

❈❛♣ít✉❧♦ ✷

❊s♣❛ç♦s ❯♥✐❢♦r♠❡s

◆❡st❡ ❝❛♣ít✉❧♦ ❢❛r❡♠♦s ✉♠❛ ❜r❡✈❡ ✐♥tr♦❞✉çã♦ ❛♦s ❡s♣❛ç♦s ✉♥✐❢♦r♠❡s ❡ ✈❡r❡♠♦s❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s r❡❧❡✈❛♥t❡s ❝♦♠♦ ❜❛s❡ ❞❡ ✉♠❛ ✉♥✐❢♦r♠✐❞❛❞❡✱ ❝❛r❛❝t❡r✐③❛çã♦❞❡ ❜❛s❡ ✉♥✐❢♦r♠✐❞❛❞❡✱ t♦♣♦❧♦❣✐❛ ✐♥❞✉③✐❞❛ ♣♦r ✉♠❛ ✉♥✐❢♦r♠✐❞❛❞❡ ❡ ✜♥❛❧♠❡♥t❡♣r♦♣r✐❡❞❛❞❡ ❞❡ ❢❡❝❤❛❞✉r❛ ♥✉♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ ✐♥❞✉③✐❞❛ ♣♦r ✉♠❛ ✉♥✐❢♦r♠✐❞❛❞❡❡ s✉❛s ❝♦♥s❡q✉ê♥❝✐❛s✳ ❬✻✱ ✹❪✳

✷✳✶ ❯♥✐❢♦r♠✐❞❛❞❡

❊♥ ✶✾✸✼ ❆♥❞ré ❲❡✐❧ ✐♥tr♦❞✉③✐✉ ❛ ♣r✐♠❡✐r❛ ❞❡✜♥✐çã♦ ❡①♣❧í❝✐t❛ ❞❡ ✉♠❛ ❡str✉t✉r❛✉♥✐❢♦r♠❡ ♦✉ ❡s♣❛ç♦ ✉♥✐❢♦r♠❡✳ ❖s ❝♦♥❝❡✐t♦s ❞❡ ✉♥✐❢♦r♠✐❞❛❞❡✱ ❡ ❝♦♠♣❧❡t✉❞❡✱ ❢♦r❛♠❞✐s❝✉t✐❞♦s ✉s❛♥❞♦ ❡s♣❛ç♦s ♠étr✐❝♦s✳ ◆✐❝♦❧❛s ❇♦✉r❜❛❦✐ ❢♦r♥❡❝❡✉ ❛ ❞❡✜♥✐çã♦ ❞❡✉♥✐❢♦r♠✐❞❛❞❡ ❡♠ t❡r♠♦s ❞❡ ✈✐③✐♥❤❛♥ç❛ ❞✐❛❣♦♥❛❧ ♥♦ ❧✐✈r♦ ✧❚♦♣♦❧♦❣✐❡ ●é♥ér❛❧❡✧❡❏♦❤♥ ❚✉❦❡② ❞❡✉ ❛ ❞❡✜♥✐çã♦ ❞❡ ❝♦❜❡rt✉r❛ ✉♥✐❢♦r♠❡✳

❉❡✜♥✐çã♦ ✷✳✶✳ ❙❡❥❛♠ X ✉♠ ❝♦♥❥✉♥t♦ ♥ã♦ ✈❛③✐♦ ❡ D = {(x, y) : (x, y) ∈ X×X}✱D−1 = {(y, x) : (x, y) ∈ D}✱ ∆ = {(x, x) : (x, x) ∈ X × X} s✉❜❝♦♥❥✉♥t♦s ❞❡X ×X✳ ❉❡✜♥❛✲s❡ ♦s s❡❣✉✐♥t❡s ❝♦♥❥✉♥t♦s ❡♠ X ×X ✿

✶✮ D é s✐♠étr✐❝♦✱ s❡ D = D−1✳

✷✮ 2D = D ◦D = {(x, z) ∈ X ×X : ∃ y ∈ X | (x, y) ∈ D ❡ (y, z) ∈ D}✳

✸✮ D é ❞✐❛❣♦♥❛❧✱ s❡ D = ∆✳

❉❡✜♥✐çã♦ ✷✳✷✳ ❙❡❥❛♠ X ✉♠ ❝♦♥❥✉♥t♦ ♥ã♦ ✈❛③✐♦✱ A ✉♠ s✉❜❝♦♥❥✉♥t♦ ❞❡ X ❡ x✉♠ ♣♦♥t♦ ❡♠ X✳ ❉❡✜♥❛✲s❡ ♦s s❡❣✉✐♥t❡s ❝♦♥❥✉♥t♦s ❡♠ X✿

✶✮ D[x] = {y ∈ X : (x, y) ∈ D}✳

✷✮ D[A] =⋃

x∈A

D[x] = {y ∈ X : ∃ x ∈ A / (x, y) ∈ D}✳

✶✷

✶✸ ✷✳✶✳ ❯◆■❋❖❘▼■❉❆❉❊

❆ ❡str✉t✉r❛ ❞❡ ✉♥✐❢♦r♠✐❞❛❞❡ ❣❡♥❡r❛❧✐③❛ ❛ ❡str✉t✉r❛ ❞♦ ❡s♣❛ç♦ ♠étr✐❝♦✱ ❛♣♦st❡r✐♦r✐ ✈❡r❡♠♦s q✉❡ t♦❞❛ ♠étr✐❝❛ ✐♥❞✉③ ✉♠❛ ✉♥✐❢♦r♠✐❞❛❞❡✳ P♦r ♦✉tr❛ ♣❛rt❡✱ ❛q✉❛❧q✉❡r ❝♦♥❥✉♥t♦ ♣♦❞❡♠♦s ♠✉♥✐✲❧♦ ❞❡ ✉♠❛ ✉♥✐❢♦r♠✐❞❛❞❡✳

❉❡✜♥✐çã♦ ✷✳✸✳ ❙❡❥❛ X ✉♠ ❝♦♥❥✉♥t♦ ♥ã♦ ✈❛③✐♦✳ ❯♠❛ ❢❛♠í❧✐❛ D ❞❡ s✉❜❝♦♥❥✉♥t♦s❞❡ X ×X é ✉♠❛ ✉♥✐❢♦r♠✐❞❛❞❡ ❡♠ X✱ s❡ s❛t✐s❢❛③ ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿

✶✮ P❛r❛ t♦❞♦ D ∈ D✱ ∆ ⊂ D✳

✷✮ ❙❡ D ∈ D✱ ❡♥tã♦ D−1 ∈ D✳

✸✮ ❙❡ D ∈ D✱ ❡♥tã♦ ❡①✐st❡ E ∈ D t❛❧ q✉❡ E ◦ E ⊆ D✳

✹✮ ❙❡ D1, D2 ∈ D✱ ❡♥tã♦ D1 ∩D2 ∈ D✳

✺✮ ❙❡ D ∈ D ❡ D ⊆ E✱ ❡♥tã♦ E ∈ D✳

❖ ♣❛r (X,D) é ❝❤❛♠❛❞♦ ❡s♣❛ç♦ ✉♥✐❢♦r♠❡✱ ♦♥❞❡ X é ✉♠ ❝♦♥❥✉♥t♦ ♥ã♦ ✈❛③✐♦ ❡D é ✉♠❛ ✉♥✐❢♦r♠✐❞❛❞❡ ❡♠ X ❡ ♦s ❡❧❡♠❡♥t♦s ❞❡ D sã♦ ❝❤❛♠❛❞♦s ❞❡ ✈✐③✐♥❤❛♥ç❛s❞✐❛❣♦♥❛❧✳

❆❣♦r❛ ❞❛q✉✐ ❡♠ ❛❞✐❛♥t❡ ❛ ♥♦t❛çã♦ ❞❡ ❡s♣❛ç♦ ✉♥✐❢♦r♠❡ só s❡rá X✱ ❛ss✉♠✐r❡♠♦sq✉❡ ✐♠♣❧✐❝✐t❛♠❡♥t❡ ❡stá ♠✉♥✐❞♦ ❞❡ ❛❧❣✉♠❛ ✉♥✐❢♦r♠✐❞❛❞❡✳ ❊♠ ♦✉tr❛s ♦❝❛s✐õ❡s s❡rá❡①♣r❡ss❛❞❛ ♣♦r (X,D) q✉❛♥❞♦ s❡ ♣r❡❝✐s❡ ❡♠ ❛❧❣✉♠❛s ❞❡✜♥✐çõ❡s ♣❛r❛ s❡✉ ♠❡❧❤♦r❡♥t❡♥❞✐♠❡♥t♦✳

◆♦ s❡❣✉✐♥t❡ ❞❡s❡♥❤♦ ♣♦❞❡♠♦s ♦❜s❡r✈❛r ❞✉❛s ✈✐③✐♥❤❛♥ç❛s ❞✐❛❣♦♥❛✐s ✐♥❞✉③✐❞❛s♣♦r ✉♠❛ ♣❛rt✐çã♦ ♥♦ ❝♦♥❥✉♥t♦ X✳

❋✐❣✉r❛ ✷✳✶✿ ❱✐③✐♥❤❛♥ç❛s ❉✐❛❣♦♥❛✐s

❆❧é♠ ❞✐ss♦✱ ♥♦ ❞❡s❡♥❤♦ t❛♠❜é♠ s❡ ♣♦❞❡ ♦❜s❡r✈❛r q✉❡ ✉♠❛ é s✉❜❝♦♥❥✉♥t♦ ❞❛♦✉tr❛✳ P❛r❛ ✐❧✉str❛r ✈❡r❡♠♦s ✉♠ ❡①❡♠♣❧♦ ❞❛ ✉♥✐❢♦r♠✐❞❛❞❡ r❡❧❛t✐✈❛✳

❊①❡♠♣❧♦ ✷✳✹✳ ❙❡❥❛ (X,D) ✉♠ ❡s♣❛ç♦ ✉♥✐❢♦r♠❡✳ ❙❡ F é ✉♠ s✉❜❝♦♥❥✉♥t♦ ❞❡ X✱❡♥tã♦✿

DF = {D ∩ (F × F ) : D ∈ D},

é ✉♠❛ ✉♥✐❢♦r♠✐❞❛❞❡ ❡♠ F ✳ ❉❡ ❢❛t♦✿

✶✸

✶✹ ✷✳✶✳ ❯◆■❋❖❘▼■❉❆❉❊

✶✮ P❛r❛ t♦❞♦ D ∈ D✱ ∆ ⊂ D ❡ ❝♦♠♦ ∆F ∈ (F × F )✱ ❡♥tã♦ ∆ ⊂ D ∩ (F × F )✱♣❛r❛ t♦❞♦ D ∩ (F × F ) ∈ DF ✳

✷✮ ❙❡❥❛♠ D ∩ (F × F ) ∈ DF ❡ D−1 ∈ D✱ ❡♥tã♦ ❝♦♠♦ D−1 ∩ (F × F ) =(D ∩ (F × F ))−1✱ ❛ss✐♠ (D ∩ (F × F )−1 ∈ DF ✳

✸✮ ❙❡❥❛♠ D ∩ (F × F ), E ∩ (F × F ) ∈ DF ❡ D ∈ D t❛❧ q✉❡ ❡①✐st❡ E ∈ D ❝♦♠E ◦E ⊆ D✱ ❡♥tã♦ E ∩ (F × F ) ∈ DF ✱ ❛ss✐♠ E ∩ (F × F ) ∩E ∩ (F × F ) ⊆D ∩ (F × F )✳

✹✮ ❙❡❥❛♠ D ∩ (F × F ), E ∩ (F × F ) ∈ DF ✱ D,E ∈ D ❡ D ∩ E ∈ D✱ ❡♥tã♦❝♦♠♦ (D ∩ E) ∩ (F × F ) = (D ∩ (F × F )) ∩ (E ∩ (F × F ))✱ ❛ss✐♠(D ∩ (F × F ) ∩ E ∩ (F × F )) ∈ DF ✳

✺✮ ❙❡❥❛♠ D ∩ (F × F ) ∈ DF ❡ D ∈ D✱ D ⊆ E t❛✐s q✉❡ E ∈ D✱ ❡♥tã♦ ❝♦♠♦D ∩ (F × F ) ⊆ E ∩ (F × F )✱ ❛ss✐♠ E ∩ (F × F ) ∈ DF ✳

P♦rt❛♥t♦ DF é ✉♠❛ ✉♥✐❢♦r♠✐❞❛❞❡ ❡♠ F ✳

❉❡✜♥✐çã♦ ✷✳✺✳ ❙❡❥❛♠ X ✉♠ ❝♦♥❥✉♥t♦ ♥ã♦ ✈❛③✐♦✱ D ✉♠❛ ✉♥✐❢♦r♠✐❞❛❞❡ ❡♠ X✳❯♠❛ ❢❛♠í❧✐❛ E ⊆ D é ❜❛s❡ ♣❛r❛ ❛ ✉♥✐❢♦r♠✐❞❛❞❡ D✱ s❡ ♣❛r❛ t♦❞♦ D ∈ D ❡①✐st❡E ∈ E t❛❧ q✉❡ E ⊆ D✳

❆ s❡❣✉✐♥t❡ ♣r♦♣♦s✐çã♦ q✉❡ ✈❛♠♦s ❡♥✉♥❝✐❛r✱ ❞✐③ q✉❡ t♦❞❛ ✉♥✐❢♦r♠✐❞❛❞❡ ♣♦ss✉✐✉♠❛ ❜❛s❡ ✉♥✐❢♦r♠✐❞❛❞❡ s✐♠étr✐❝❛✳

Pr♦♣♦s✐çã♦ ✷✳✻✳ ❙❡ (X,D) ✉♠ ❡s♣❛ç♦ ✉♥✐❢♦r♠❡✱ ❡♥tã♦ ❛ ❢❛♠í❧✐❛

S(D) = {D ∈ D : D = D−1}

é ❜❛s❡ ♣❛r❛ ❛ ✉♥✐❢♦r♠✐❞❛❞❡ D ❡♠ X✳

❉❡♠♦♥str❛çã♦✳ ❙❡❥❛ D ✉♠❛ ✉♥✐❢♦r♠✐❞❛❞❡ ❡♠ X✳ ❖❜s❡r✈❡ q✉❡ ♣❛r❛ t♦❞♦ D ∈ D✱❡①✐st❡ E = D ∩D−1 ∈ S(D)✱ ♣♦✐s

E−1 = (D ∩D−1)−1 = (D−1)−1 ∩D−1 = D ∩D−1 = E.

❆ss✐♠✱ E = E−1✳ ❈♦♠♦ E = D ∩D−1✱ ❡♥tã♦ E = D ∩D−1 ⊆ D✳ ▲♦❣♦✱ S(D) é✉♠❛ ❜❛s❡ ♣❛r❛ ❛ ✉♥✐❢♦r♠✐❞❛❞❡ D ❡♠ X✳

❉❡✜♥✐çã♦ ✷✳✼✳ ❙❡❥❛ (X,D) ✉♠ ❡s♣❛ç♦ ✉♥✐❢♦r♠❡✳ ❯♠ s✉❜❝♦♥❥✉♥t♦ S ❞❡ D é ✉♠❛s✉❜✲❜❛s❡ ❞❡ D✱ s❡ t♦❞❛s ❛s ✐♥t❡rs❡❝çõ❡s ✜♥✐t❛s ❞❡ ❡❧❡♠❡♥t♦ ❞❡ E ❢♦r♠❛♠ ✉♠❛❜❛s❡ ♣❛r❛ ❛ ✉♥✐❢♦r♠✐❞❛❞❡ D✳

❆❣♦r❛✱ ❞❛r❡♠♦s ✉♠ ❡①❡♠♣❧♦ ✐♠♣♦rt❛♥t❡ ❞❡ ✉♠❛ s✉❜✲❜❛s❡ ♣❛r❛ ✉♠❛✉♥✐❢♦r♠✐❞❛❞❡ ♥✉♠ ♣r♦❞✉t♦✳

❖❜s❡r✈❛çã♦ ✷✳✽✳ ❈♦♥s✐❞❡r❡✲s❡ X1, X2, ..., Xn, ... ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❡s♣❛ç♦s✉♥✐❢♦r♠❡s ❝♦♠ ❛s ✉♥✐❢♦r♠✐❞❛❞❡s D1,D2, ...,Dn, ... r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❊♥tã♦✱ ❞❡✜♥❛✲s❡ ✉♠❛ s✉❜✲❜❛s❡ ✉♥✐❢♦r♠✐❞❛❞❡ E ❡♠

∏

n∈N

Xn✳

✶✹

✶✺ ✷✳✷✳ ❇❆❙❊ ❯◆■❋❖❘▼■❉❆❉❊

P❛r❛ t♦❞♦ n ∈ N ❡ ♣❛r❛ t♦❞♦ D ∈ D ❛ s✉❜✲❜❛s❡ ✉♥✐❢♦r♠✐❞❛❞❡∏

n∈N

Xn ❡st❛ ❢♦r♠❛❞♦

♣❡❧♦s ❡❧❡♠❡♥t♦ ❞❛ ❢♦r♠❛✿

E = {(x, y) ∈∏

n∈N

Xn ×∏

n∈N

Xn : (x(n), y(n)) ∈ D}.

❆❣♦r❛✱ ♦❜s❡r✈❡ q✉❡ ❛ ✐♥t❡rs❡çã♦ ✜♥✐t❛ ❞❡ ❡❧❡♠❡♥t♦ ❞❡ E ❢♦r♠❛ ✉♠❛ ❜❛s❡ ✉♥✐❢♦r♠❡♣❛r❛ ❛ ✉♥✐❢♦r♠✐❞❛❞❡ ❡♠

∏

n∈N

Xn✳

✷✳✷ ❇❛s❡ ❯♥✐❢♦r♠✐❞❛❞❡

◆❡st❛ s❡çã♦✱ ❛ ✐❞❡✐❛ ♦❜t❡r ♠ét♦❞♦s ♣❛r❛ ❝♦♥str✉✐r ✉♥✐❢♦r♠✐❞❛❞❡ ♥✉♠ ❝♦♥❥✉♥t♦❳ ❛ ♣❛rt✐r ❞❡ s✉❜✲❝♦❧❡çõ❡s ❞❡ s✉❜❝♦♥❥✉♥t♦s ❞❡ X × X✱ q✉❡ s❛t✐s❢❛③❡♠ ❝❡rt❛s♣r♦♣r✐❡❞❛❞❡s ♣r❡❡st❛❜❡❧❡❝✐❞❛s✳ ❆ ❞❡✜♥✐çã♦ ❞❡ ❜❛s❡ ✉♥✐❢♦r♠✐❞❛❞❡ é ✉♠ ♠ét♦❞♦♣❛r❛ ❣❡r❛r ✉♠❛ ✉♥✐❢♦r♠✐❞❛❞❡ ♥✉♠ ❝♦♥❥✉♥t♦✱ ✉t✐❧✐③❛♥❞♦ ❛ ❝♦❧❡çã♦ ❞❡ s✉❜❝♦♥❥✉♥t♦s❞❡ X ×X✱ ❞❡ t❛❧ ❢♦r♠❛ q✉❡ ❡❧❛ r❡s✉❧t❡ s❡r ✉♠❛ ❜❛s❡ ✉♥✐❢♦r♠❡ ❞❛ ✉♥✐❢♦r♠✐❞❛❞❡❣❡r❛❞❛✳

❉❡✜♥✐çã♦ ✷✳✾✳ ❙❡❥❛ X ✉♠ ❝♦♥❥✉♥t♦ ♥ã♦ ✈❛③✐♦✳ ❯♠❛ ❢❛♠í❧✐❛ E ❞❡ s✉❜❝♦♥❥✉♥t♦s❞❡ X×X é ✉♠❛ ❜❛s❡ ✉♥✐❢♦r♠✐❞❛❞❡ ❡♠ X✱ s❡ s❛t✐s❢❛③ ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿

✶✮ P❛r❛ t♦❞♦ E ∈ E✱ ∆ ⊂ E✳

✷✮ ❙❡ E ∈ E✱ ❡♥tã♦ ❡①✐st❡ F ∈ E t❛❧ q✉❡ F−1 ⊆ E✳

✸✮ ❙❡ E ∈ E✱ ❡♥tã♦ ❡①✐st❡ F ∈ E t❛❧ q✉❡ F ◦ F ⊆ E✳

✹✮ ❙❡ E1, E2 ∈ E✱ ❡♥tã♦ E1 ∩ E2 ∈ E✳

✺✮ ❙❡ E ∈ E ❡ E ⊆ F ✱ ❡♥tã♦ F ∈ E✳

❚❡♥❞♦ ❞❡✜♥✐❞❛ ❜❛s❡ ✉♥✐❢♦r♠✐❞❛❞❡ ❛❣♦r❛ ✈❡r❡♠♦s ❛❧❣✉♥s ❡①❡♠♣❧♦ ❞❡st❡s✱ ♣♦✐ss❡rã♦ ❞❡ ♠✉✐t❛ ✐♠♣♦rtâ♥❝✐❛ ♠❛✐s ❛❞✐❛♥t❡✳

❊①❡♠♣❧♦ ✷✳✶✵✳ ❙❡❥❛ S ❛ ❧✐♥❤❛ ❞❡ ❙♦r❣❡♥❢r❡②✳ ❆ ❢❛♠í❧✐❛ D ❞❡ s✉❜❝♦♥❥✉♥t♦s ❞❡S × S ❢♦r♠❛❞♦ ♣❡❧♦s ❡❧❡♠❡♥t♦s ❞❛ ❢♦r♠❛✿

D = [a, b)× [a, b)

é ✉♠❛ ❜❛s❡ ✉♥✐❢♦r♠✐❞❛❞❡ ❡♠ S✱ ✐ss♦ q✉❡r ❞✐③❡r q✉❡ ❡①✐st❡ ✉♠❛ ✉♥✐❢♦r♠✐❞❛❞❡ q✉❡❣❡r❛ ❛ t♦♣♦❧♦❣✐❛ ❞❡ ❙♦r❣❡♥❢r❡②✳

❊①❡♠♣❧♦ ✷✳✶✶✳ ❙❡❥❛X ✉♠ ❝♦♥❥✉♥t♦ ♥ã♦ ❡♥✉♠❡rá✈❡❧✳ ❆ ❢❛♠í❧✐❛D ❞❡ s✉❜❝♦♥❥✉♥t♦❞❡ X ×X✱ ❢♦r♠❛❞♦ ♣❡❧♦s ❡❧❡♠❡♥t♦s ❞❛ ❢♦r♠❛✿

DF = F × F ∪ (X − F )× (X − F ),

é ✉♠❛ ❜❛s❡ ✉♥✐❢♦r♠✐❞❛❞❡ ❡♠ X✱ ♦♥❞❡ F é ✉♠ s✉❜❝♦♥❥✉♥t♦ ✜♥✐t♦ ❞❡ X✳

✶✺

✶✻ ✷✳✷✳ ❇❆❙❊ ❯◆■❋❖❘▼■❉❆❉❊

◆❛ ♣r♦♣♦s✐çã♦ q✉❡ ♠❡♥❝✐♦♥❛r❡♠♦s ❛ s❡❣✉✐r✱ ❝❛r❛❝t❡r✐③❛r❡♠♦s ❜❛s❡ ❞❡ ✉♠❛✉♥✐❢♦r♠✐❞❛❞❡✳

Pr♦♣♦s✐çã♦ ✷✳✶✷✳ ❙❡❥❛♠ X ✉♠ ❝♦♥❥✉♥t♦ ♥ã♦ ✈❛③✐♦ ❡ E ✉♠❛ ❢❛♠í❧✐❛ ♥ã♦ ✈❛③✐❛❞❡ s✉❜❝♦♥❥✉♥t♦ ❞❡ X ×X✳ P♦❞❡♠♦s ❛✜r♠❛r q✉❡✿

✶✮ ❙❡ E é ✉♠❛ ❜❛s❡ ✉♥✐❢♦r♠✐❞❛❞❡ ❡♠ X✱ ❡♥tã♦ DE = {D ⊂ X × X : ∃E ∈E, E ⊆ D} é ✉♠❛ ✉♥✐❢♦r♠✐❞❛❞❡ ❡♠ X✳

✷✮ ❙❡ D é ✉♠❛ ✉♥✐❢♦r♠✐❞❛❞❡ ❡♠ X✱ E ⊆ D ❡ D = DE✱ ❡♥tã♦ E é ✉♠❛ ❜❛s❡✉♥✐❢♦r♠✐❞❛❞❡ ❡♠ X✳

❉❡♠♦♥str❛çã♦✳ ▼♦str❡♠♦s q✉❡ DE = {D ⊂ X × X : ∃E ∈ E, E ⊆ D} é ✉♠❛✉♥✐❢♦r♠✐❞❛❞❡ ❡♠ X✱ ♦✉ s❡❥❛✱ DE s❛t✐s❢❛③ ❛s ♣r♦♣r✐❡❞❛❞❡s ❞❛ ❉❡✜♥✐çã♦ ✷✳✸✳

✶✮ ❙❡ D ∈ DE✱ ❡♥tã♦ ❡①✐st❡ E ∈ E t❛❧ q✉❡ E ⊆ D✱ ❡ ❝♦♠♦ ∆ ⊆ E✱ ❡♥tã♦∆ ⊆ D✳

✷✮ ❙❡ D ∈ DE✱ ❡♥tã♦ ❡①✐st❡ E ∈ E t❛❧ q✉❡ E ⊆ D✱ ❡ ❝♦♠♦ E é ♠❛❜❛s❡ ✉♥✐❢♦r♠✐❞❛❞❡ ❡①✐st❡ F ∈ E t❛❧ q✉❡ F−1 ⊆ E✱ ♦ q✉❡ ✐♠♣❧✐❝❛ q✉❡F ⊆ E−1 ⊆ D−1 ❛ss✐♠ D−1 ∈ DE✳

✸✮ ❙❡ D ∈ DE✱ ❡♥tã♦ ❡①✐st❡ E ∈ E t❛❧ q✉❡ E ⊆ D✱ ❡ ❝♦♠♦ E é ♠❛ ❜❛s❡✉♥✐❢♦r♠✐❞❛❞❡ ❡①✐st❡ F ∈ E t❛❧ q✉❡ F ◦ F ⊆ E ❛ss✐♠ F ◦ F ⊆ D✳

✹✮ ❙❡ D1, D2 ∈ DE✱ ❡♥tã♦ ❡①✐st❡♠ E1, E2 ∈ E t❛✐s q✉❡ E1 ⊆ D1 ❡ E2 ⊆ D2✳❊①✐st❡ E3 ∈ E t❛❧ q✉❡ E3 ⊆ E1 ∩ E2 ⊆ D1 ∩D2✱ ❛ss✐♠ D1 ∩D2 ∈ DE✳

✺✮ ❙❡ E ∈ DE ❡ E ⊆ D✱ ❡♥tã♦ ❡①✐st❡ E′

∈ E t❛❧ q✉❡ E′

⊆ E ⊆ D✱ ❛ss✐♠D ∈ DE✳

P♦rt❛♥t♦✱ DE é ✉♠❛ ✉♥✐❢♦r♠✐❞❛❞❡ ❡♠ X✳ ❆❣♦r❛✱ ♠♦str❡♠♦s q✉❡ E é ✉♠❛ ❜❛s❡✉♥✐❢♦r♠✐❞❛❞❡ ❡♠ X✱ ♦✉ s❡❥❛✱ E s❛t✐s❢❛③ ❛s ♣r♦♣r✐❡❞❛❞❡s ❞❛ ❉❡✜♥✐çã♦ ✷✳✾✳

✶✮ ❙❡ E ∈ E✱ ❡♥tã♦ E ∈ D✱ ∆ ⊆ E✳

✷✮ ❙❡ E ∈ E✱ ❡♥tã♦ E ∈ D✱ ♦ q✉❡ ✐♠♣❧✐❝❛ q✉❡ E−1 ∈ D✳ ❊①✐st❡ F ∈ E t❛❧ q✉❡F ⊆ E−1✱ ❛ss✐♠ F−1 ⊆ E✳

✸✮ ❙❡ E ∈ E✱ ❡♥tã♦ E ∈ D✱ ❡①✐st❡ F ∈ D t❛❧ q✉❡ F ◦ F ⊆ E✳ P♦r ♦✉tr♦ ❧❛❞♦❡①✐st❡ H ∈ E t❛❧ q✉❡ H ⊆ F ✳ ❆ss✐♠ H ◦H ⊆ F ◦ F ⊆ E✳

✹✮ ❙❡ E1, E2 ∈ E✱ ❡♥tã♦ E1, E2 ∈ D ♦ q✉❡ ✐♠♣❧✐❝❛ q✉❡ E1 ∩ E2 ∈ D✳ ❚♦♠❡E1 ∩ E2 = E3✱ t❡♠♦s E3 ⊆ E1 ∩ E2✳

✺✮ ❙❡ D ∈ E ❡ D ⊆ E✱ ❡♥tã♦ D ∈ D✱ ❛ss✐♠ E ∈ E✳

P♦rt❛♥t♦✱ E é ✉♠❛ ❜❛s❡ ✉♥✐❢♦r♠✐❞❛❞❡ ❡♠ X✳

✶✻

✶✼ ✷✳✷✳ ❇❆❙❊ ❯◆■❋❖❘▼■❉❆❉❊

❆ ❝♦♥t✐♥✉❛çã♦ ✈❡r❡♠♦s ✉♠ ❡①❡♠♣❧♦ ❞❡ ✉♠❛ ❜❛s❡ ✉♥✐❢♦r♠✐❞❛❞❡ ✐♥❞✉③✐❞❛ ♣♦r✉♠❛ ♠étr✐❝❛✳

Pr♦♣♦s✐çã♦ ✷✳✶✸✳ ❙❡ (X, d) é ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦ ❡ E ✉♠❛ ❢❛♠í❧✐❛ ❞❡s✉❜❝♦♥❥✉♥t♦s ❞❡ X ×X✱ ❢♦r♠❛❞♦ ♣❡❧♦s ❡❧❡♠❡♥t♦s ❞❛ ❢♦r♠❛✿

En = {(x, y) ∈ X ×X : d(x, y) < 2−n}.

❊♥tã♦ E é ✉♠❛ ❜❛s❡ ✉♥✐❢♦r♠✐❞❛❞❡ ❡♠ X✳

❉❡♠♦♥str❛çã♦✳ ❙❡❥❛♠ (X, d) ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦ ❡ n ∈ N✳ ▼♦str❡♠♦s q✉❡ E é✉♠❛ ❜❛s❡ ✉♥✐❢♦r♠✐❞❛❞❡ ❡♠ X✳ ❉❡ ❢❛t♦✿

✶✮ ❙❡❥❛ En ∈ E✱ ♣❛r❛ t♦❞♦ n ∈ N t❡♠✲s❡ q✉❡✿

(x, x) ∈ ∆ ⇐⇒ d(x, x) = 0 < 2−n

=⇒ (x, x) ∈ En.

❆ss✐♠ ∆ ⊆ En ♣❛r❛ t♦❞♦ n ∈ N

✷✮ ❙❡❥❛ En ∈ E✱ ♣❛r❛ t♦❞♦ n ∈ N ❡①✐st❡ En+1 ∈ E t❛❧ q✉❡✿

(y, x) ∈ E−1n+1 =⇒ d(y, x) = d(x, y) < 2−(n+1) < 2−n

=⇒ (y, x) ∈ En.

▲♦❣♦ E−1n+1 ⊆ En✳

✸✮ ❙❡❥❛ En ∈ E✱ ♣❛r❛ t♦❞♦ n ∈ N ❡①✐st❡ En+1 ∈ E t❛❧ q✉❡✿

(x, z) ∈ En+1 ◦ En+1 =⇒ ∃ y ∈ X t❛❧ q✉❡ (x, y) ∈ En+1 ❡ (y, z) ∈ En+1

⇐⇒ d(x, y) < 2−(n+1) ❡ d(y, z) < 2−(n+1)

=⇒ d(x, z) ≤ d(x, y) + d(y, z) < 2−n

=⇒ (x, z) ∈ En.

❆ss✐♠ En+1 ◦ En+1 ⊆ En✳

✹✮ ❙❡❥❛♠ En, Em ∈ E✳ ❊♥tã♦ En ⊆ Em ♦✉ Em ⊆ En✳ ❙✉♣♦♥❤❛♠♦s q✉❡En ⊆ Em ❡♥tã♦ En ⊂ En ∩ Em✱ ❛ss✐♠ En ∩ Em ∈ E✳

✺✮ ❙❡❥❛♠ En ∈ E ❡ En ⊆ Em✳ ❊♥tã♦ t❡♠✲s❡✿

(x, y) ∈ En ⇐⇒ d(x, y) < 2n < 2m

=⇒ (x, y) ∈ Em.

▲♦❣♦ Em ∈ E✳

P♦rt❛♥t♦✱ E é ✉♠❛ ❜❛s❡ ✉♥✐❢♦r♠✐❞❛❞❡ ❡♠ X✳

✶✼

✶✽ ✷✳✸✳ P❘❖P❘■❊❉❆❉❊❙ ❉❊ ❯◆■❋❖❘▼■❉❆❉❊

✷✳✸ Pr♦♣r✐❡❞❛❞❡s ❞❡ ❯♥✐❢♦r♠✐❞❛❞❡

❆ ✉♥✐❢♦r♠✐❞❛❞❡ ❛♣r❡s❡♥t❛ ✉♠❛ ❡str✉t✉r❛ ♠❛✐s ❢♦rt❡ q✉❡ ✉♠❛ t♦♣♦❧♦❣✐❛✱ ♣❡❧♦q✉❛❧ t♦❞❛ ✉♥✐❢♦r♠✐❞❛❞❡ ✐♥❞✉③ ✉♠❛ t♦♣♦❧♦❣✐❛✳ ❆ ❝♦♥t✐♥✉❛çã♦ ❞❡✜♥✐r❡♠♦s ✉♠❛t♦♣♦❧♦❣✐❛ ✐♥❞✉③✐❞❛ ♣♦r ✉♠❛ ✉♥✐❢♦r♠✐❞❛❞❡✱ ♣❛r❛ ♦ q✉❛❧ ♣r❡❝✐s❛♠♦s ♦ s❡❣✉✐♥t❡t❡♦r❡♠❛✳

❚❡♦r❡♠❛ ✷✳✶✹✳ ❙❡ (X,D) é ✉♠ ❡s♣❛ç♦ ✉♥✐❢♦r♠❡ ❡ x é ✉♠ ❡❧❡♠❡♥t♦ ❞❡ X✱ ❡♥tã♦V(x) = {D[x] : D ∈ D} é ✉♠❛ ❜❛s❡ ❞❡ ✈✐③✐♥❤❛♥ç❛s ❞❡ x✳

❉❡♠♦♥str❛çã♦✳ ❙❡❥❛ (X,D) é ✉♠ ❡s♣❛ç♦ ✉♥✐❢♦r♠❡ ❡ x ✉♠ ❡❧❡♠❡♥t♦ ❡♠ X✱♠♦str❡♠♦s q✉❡ V(x) = {D[x] : D ∈ D} é ✉♠❛ ❜❛s❡ ❞❡ ✈✐③✐♥❤❛♥ç❛s ❡♠ x✳ Pr✐♠❡✐r♦♦❜s❡r✈❡ q✉❡ ♣❛r❛ t♦❞♦ E ∈ E✱ (x, x) ∈ ∆ ⊆ E✱ ♦ q✉❡ ✐♠♣❧✐❝❛ q✉❡ x ∈ E[x]✳ ❈♦♠✐ss♦ t❡♠♦s✿

✶✮ ❙❡ D1[x], D2[x] ∈ V(x)✱ ❡♥tã♦ D1, D2 ∈ D ❡ D1 ∩D2 ∈ D✳ ▲♦❣♦✿

y ∈ D1[x] ∩D2[x] ⇐⇒ y ∈ D1[x] ❡ y ∈ D2[x]

⇐⇒ (x, y) ∈ D1 ❡ (x, y) ∈ D2

⇐⇒ (x, y) ∈ (D1 ∩D2)

⇐⇒ y ∈ (D1 ∩D2)[x],

❛ss✐♠ D1[x] ∩D2[x] ∈ V(x)✳

✷✮ ❙❡ D[x] ∈ V(x)✱ ❡♥tã♦ D ∈ D✳ ▲♦❣♦✳

D ∈ D =⇒ ∃ E ∈ D t❛❧ q✉❡ E ◦ E ⊆ D

=⇒ E ◦ E[x] ∈ V(x).

P♦r ♦✉tr♦ ❧❛❞♦ s❡❥❛ y ∈ E[x] ❡ z ∈ E[y] ∈ V(y)✱

y ∈ E[x] ❡ z ∈ E[y] =⇒ (x, y) ∈ E ❡ (y, z) ∈ E

=⇒ (x, z) ∈ 2E ⊂ D

=⇒ z ∈ D[x].

❆ss✐♠ E[y] ⊆ D[x]✳

P♦rt❛♥t♦✱ V(x) é ✉♠❛ ❜❛s❡ ❞❡ ✈✐③✐♥❤❛♥ç❛s ❡♠ x✳

Pr♦♣♦s✐çã♦ ✷✳✶✺✳ ❙❡❥❛ (X,D) ✉♠ ❡s♣❛ç♦ ✉♥✐❢♦r♠❡ ❡ x ∈ X✳ ❙❡ E é ❜❛s❡ ❞❛✉♥✐❢♦r♠✐❞❛❞❡ D✱ ❡♥tã♦ B(x) = {E[x] : E ∈ E} é ✉♠❛ ❜❛s❡ ❞❡ ✈✐③✐♥❤❛♥ç❛s ❡♠ x✳

❉❡♠♦♥str❛çã♦✳ ❙❡❥❛♠ (X,D) ✉♠ ❡s♣❛ç♦ ✉♥✐❢♦r♠❡✱ x ✉♠ ❡❧❡♠❡♥t♦ ❡♠ X ❡ E ✉♠❛❜❛s❡ ♣❛r❛ ❛ ✉♥✐❢♦r♠✐❞❛❞❡ D✳ ▼♦str❡♠♦s q✉❡ B(x) = {E[x] : E ∈ E} é ✉♠❛ ❜❛s❡❞❡ ✈✐③✐♥❤❛♥ç❛s ❡♠ x✳ Pr✐♠❡✐r♦ ♦❜s❡r✈❡ q✉❡ ♣❛r❛ t♦❞♦ E ∈ E✱ (x, x) ∈ ∆ ⊆ E✱ ♦q✉❡ ✐♠♣❧✐❝❛ q✉❡ x ∈ E[x]✳ ❈♦♠ ✐ss♦ t❡♠♦s✿

✶✽

✶✾ ✷✳✸✳ P❘❖P❘■❊❉❆❉❊❙ ❉❊ ❯◆■❋❖❘▼■❉❆❉❊

✶✮ ❙❡ D1[x], D2[x] ∈ B(x)✱ ❡♥tã♦ E1, E2 ∈ E ❡ E1 ∩ E2 ∈ E✳ ▲♦❣♦✿

y ∈ E1[x] ∩ E2[x] ⇐⇒ y ∈ E1[x] ❡ y ∈ E2[x]

⇐⇒ (x, y) ∈ E1 ❡ (x, y) ∈ E2

⇐⇒ (x, y) ∈ (E1 ∩ E2)

⇐⇒ y ∈ (E1 ∩ E2)[x].

❆ss✐♠ E1[x] ∩ E2[x] ∈ B(x)✳

✷✮ ❙❡ E[x] ∈ B(x)✱ ❡♥tã♦ E ∈ E✳ ▲♦❣♦✱

E ∈ E =⇒ ∃F ∈ E : F ◦ F ⊆ E

=⇒ F ◦ F [x] ∈ B(x).

P♦r ♦✉tr♦ ❧❛❞♦ s❡❥❛ y ∈ F [x] ❡ z ∈ F [y] ∈ B(y)

y ∈ F [x] ❡ z ∈ F [y] =⇒ (x, y) ∈ F ❡ (y, z) ∈ F

=⇒ (x, z) ∈ 2F ⊆ E

=⇒ z ∈ E[x],

❛ss✐♠ F [y] ⊆ E[x]✳

P♦rt❛♥t♦✱ B(x) é ✉♠❛ ❜❛s❡ ❞❡ ✈✐③✐♥❤❛♥ç❛s ❡♠ x✳

❆❣♦r❛ ❝♦♠♦ ❝♦♥s❡q✉ê♥❝✐❛ ❞❛ Pr♦♣♦s✐çã♦ ✷✳✶✺✱ t❡♥❞♦ ✉♠❛ ❜❛s❡ ❞❡ ✈✐③✐♥❤❛♥ç❛❧♦❝❛❧ ❡♠ ❢✉♥çã♦ ❞❡ ✈✐③✐♥❤❛♥ç❛s ❞✐❛❣♦♥❛✐s✱ ❡♥tã♦ ♣♦❞❡♠♦s ❣❡r❛r ✉♠❛ t♦♣♦❧♦❣✐❛❝❤❛♠❛❞❛ t♦♣♦❧♦❣✐❛ ✉♥✐❢♦r♠❡✳

❉❡✜♥✐çã♦ ✷✳✶✻✳ ❙❡❥❛ (X,D) ✉♠ ❡s♣❛ç♦ ✉♥✐❢♦r♠❡✳ ❆ t♦♣♦❧♦❣✐❛ ✉♥✐❢♦r♠❡ ❡♠ X✱✐♥❞✉③✐❞❛ ♣❡❧❛ ✉♥✐❢♦r♠✐❞❛❞❡ D é ❞❛❞❛ ♣♦r✿

TD = {U ⊆ X : ∀ x ∈ U, ∃ D ∈ D; D[x] ⊆ U}.

❖ ♣❛r (X,TD) é ❝❤❛♠❛❞♦ ❞❡ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ ✉♥✐❢♦r♠❡✱ ♦♥❞❡ TD é ❛t♦♣♦❧♦❣✐❛ ❣❡r❛❞❛ ♣❡❧❛ ✉♥✐❢♦r♠✐❞❛❞❡ D✳

❖❜s❡r✈❛çã♦ ✷✳✶✼✳ ❙❡❥❛ (X,D) ✉♠ ❡s♣❛ç♦ ✉♥✐❢♦r♠❡✳ D ∈ D é ❛❜❡rt♦✱ s❡ D é❛❜❡rt♦ ❡♠ X ×X ❝♦♠ ❛ t♦♣♦❧♦❣✐❛ ♣r♦❞✉t♦✳ P♦r ♦✉tr❛ ♣❛rt❡ t❡♠✲s❡ q✉❡ D ∈ D

é ❛❜❡rt♦✱ ❡♥tã♦ D[x] é ❛❜❡rt♦ ❡♠ X ♣❛r❛ t♦❞♦ x ∈ X✳

❉❡s❞❡ q✉❡ t♦❞❛ ✉♥✐❢♦r♠✐❞❛❞❡ ✐♥❞✉③ ✉♠❛ t♦♣♦❧♦❣✐❛✱ ♣❛ss❛♠♦s ❛ ❞❡✜♥✐r ✉♠❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ ✉♥✐❢♦r♠✐③❛✈é❧✳

❉❡✜♥✐çã♦ ✷✳✶✽✳ ❯♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ X é ✉♥✐❢♦r♠✐③á✈❡❧✱ s❡ ❡①✐st❡ ✉♠❛✉♥✐❢♦r♠✐❞❛❞❡ D q✉❡ ✐♥❞✉③ ❛ t♦♣♦❧♦❣✐❛ ❡♠ X✳

❊①❡♠♣❧♦ ✷✳✶✾✳ ❙❡❥❛ R ❛ r❡t❛ r❡❛❧✳ ❈♦♥s✐❞❡r❡ ❛ ❢❛♠í❧✐❛ D ❞❡ s✉❜❝♦♥❥✉♥t♦ ❞❡R× R✱ ❢♦r♠❛❞❛ ♣❡❧♦s ❡❧❡♠❡♥t♦s ❞❛ ❢♦r♠❛✿

Dε = {(x, y) ∈ R× R : |x− y| < ε}.

✶✾

✷✵ ✷✳✸✳ P❘❖P❘■❊❉❆❉❊❙ ❉❊ ❯◆■❋❖❘▼■❉❆❉❊

é ✉♠❛ ❜❛s❡ ✉♥✐❢♦r♠✐❞❛❞❡ R✳ ❊st❛ ❜❛s❡ ✉♥✐❢♦r♠✐❞❛❞❡ D ✐♥❞✉③ ✉♠❛ ❜❛s❡ t♦♣♦❧ó❣✐❝❛❛ q✉❛❧ ❣❡r❛ ❛ t♦♣♦❧♦❣✐❛ ✉s✉❛❧ ❡♠ R✳

❚❡♦r❡♠❛ ✷✳✷✵✳ ❙❡❥❛ (X,D) ✉♠ ❡s♣❛ç♦ ✉♥✐❢♦r♠❡✱ (X,TD) ✉♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦❡ A ✉♠ s✉❜❝♦♥❥✉♥t♦ ❞❡ X✳ ❙❡ E é ❜❛s❡ ❞❛ ✉♥✐❢♦r♠✐❞❛❞❡ D✱ ❡♥tã♦

A =⋂

E∈E

E[A].

❉❡♠♦♥str❛çã♦✳ ❙❡❥❛♠ x ∈ A✱ E ∈ E ❡ F ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❞✐❛❣♦♥❛❧ s✐♠étr✐❝❛ ❞❡E t❛❧ q✉❡ F ⊆ E✳ ❊♥tã♦ F [x] é ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❞❡ x ❡ A∩F [x] 6= ∅✳ ▲♦❣♦ ❡①✐st❡a ∈ A ∩ F [x] t❛❧ q✉❡✿

(x, a) ∈ F =⇒ (a, x) ∈ F

=⇒ x ∈ F [a] ⊆ F [A] ⊆ E[A]

=⇒ ∀E ∈ E, x ∈ E[A].

❆ss✐♠ x ∈⋂

E∈E

E[A]✳

❙❡❥❛♠ x ∈⋂

E∈E

E[A] ❡ V[x] ✉♠❛ ❜❛s❡ ❞❡ ✈✐③✐♥❤❛♥ç❛s s✐♠étr✐❝❛s ❞❡ x✳ ❊♥tã♦✿

x ∈ E[A] =⇒ ∃ a ∈ A, t❛❧ q✉❡ (a, x) ∈ E

=⇒ (x, a) ∈ E ❡ a ∈ E[x]

=⇒ a ∈ E[x] ∩ A

=⇒ E[x] ∩ A 6= ∅.

▲♦❣♦ x ∈ A✳ P♦rt❛♥t♦✱ A =⋂

E∈E

E[A]✳

❆♥t❡s ❞❡ ❛♣r❡s❡♥t❛r ♦ ♣ró①✐♠♦ ❧❡♠❛✱ ♣r❡❝✐s❛♠♦s ❞❛ s❡❣✉✐♥t❡ ♦❜s❡r✈❛çã♦✳

❖❜s❡r✈❛çã♦ ✷✳✷✶✳ ❙❡ X é ✉♠ ❡s♣❛ç♦ ✉♥✐❢♦r♠❡✱ ♦❜s❡r✈❡ q✉❡ D[x] ⊆ 2D[x] ♣❛r❛t♦❞♦ D ♥❛ ✉♥✐❢♦r♠✐❞❛❞❡✳ ❉❡ ❢❛t♦✱ ♣♦r ❞❡✜♥✐çã♦ t❡♠✲s❡✿

D[x] =⋂

E∈E

E[D[x]]

⊆ D[D[x]] = 2D[x].

❆ss✐♠ D[x] ⊆ 2D[x] ♣❛r❛ t♦❞♦ D ❡ x ∈ X✳

▲❡♠❛ ✷✳✷✷✳ ❙❡❥❛♠ (X,D) ✉♠ ❡s♣❛ç♦ ✉♥✐❢♦r♠❡✱ D ∈ D s✐♠étr✐❝♦ ❡ x ∈ X✳ ❙❡A ⊆ D[x] ❡ y ∈ A✱ ❡♥tã♦ A ⊆ 4D[y]✳

❉❡♠♦♥str❛çã♦✳ ❙❡❥❛ D ∈ D s✐♠étr✐❝♦✱ y ∈ A ❡ A ⊆ D[x]✳ ❊♥tã♦✱ A ⊆ D[x] ⊆2D[x]✱ ♦ q✉❡ ✐♠♣❧✐❝❛ q✉❡ y ∈ 2D[x] ❡ (x, y) ∈ 2D[x]✳ P♦r ♦✉tr❛ ♣❛rt❡✱ ♣❛r❛ t♦❞♦

✷✵

✷✶ ✷✳✸✳ P❘❖P❘■❊❉❆❉❊❙ ❉❊ ❯◆■❋❖❘▼■❉❆❉❊

z ∈ A t❡♠✲s❡✿

z ∈ A ⊆ D[x] ⊆ 2D[x] =⇒ (z, x) ∈ 2D

=⇒ (z, x) ∈ 2D ❡ (x, y) ∈ 2D

=⇒ (z, y) ∈ 4D =⇒ (y, z) ∈ 4D

=⇒ z ∈ 4D[y].

P♦rt❛♥t♦✱ A ⊆ 4D[y]✳

❆❣♦r❛ ✜♥❛❧♠❡♥t❡ ❡♥✉♥❝✐❛r❡♠♦s ❛s s❡❣✉✐♥t❡s ❞❡✜♥✐çõ❡s ❞❡ ❝♦❜❡rt✉r❛ ♥✉♠❡s♣❛ç♦ ✉♥✐❢♦r♠❡ ❡ r❡✜♥❛♠❡♥t♦ ❡str❡❧❛✳

❉❡✜♥✐çã♦ ✷✳✷✸✳ ❙❡❥❛♠ (X,D) ✉♠ ❡s♣❛ç♦ ✉♥✐❢♦r♠❡✱ U ❡ V ❝♦❜❡rt✉r❛s ❞❡ X ❡ A✉♠ s✉❜❝♦♥❥✉♥t♦ ❞❡ X✳ ❉✐③❡♠♦s q✉❡✿

✶✮ U é ✉♠❛ ❝♦❜❡rt✉r❛ ✉♥✐❢♦r♠❡✱ s❡ ❡①✐st❡ D ∈ D t❛❧ q✉❡ UD = {D[x] : x ∈X} r❡✜♥❛ ❛ U✳

✷✮ ❯♠❛ ❡str❡❧❛ ❞♦ ❝♦♥❥✉♥t♦ A✱ ❝♦♠ r❡s♣❡✐t♦ à ❝♦❜❡rt✉r❛ ✉♥✐❢♦r♠❡ U✱ é ♦❝♦♥❥✉♥t♦ S(A,U) =

⋃

U∈U′

U ✱ ♦♥❞❡ U′

= {U ∈ U : U ∩ A 6= ∅}✳

✸✮ U é ✉♠ r❡✜♥❛♠❡♥t♦ ❡str❡❧❛ ❞❡ V✱ s❡ ♣❛r❛ t♦❞♦ U ∈ U✱ ❡①✐st❡ V ∈ V t❛❧q✉❡ S(U,U) ⊂ V ✳

✹✮ U é ✉♠❛ ❝♦❜❡rt✉r❛ ♥♦r♠❛❧✱ s❡ U é ✉♠❛ ❝♦❜❡rt✉r❛ ❛❜❡rt❛ ❡ s❡ ❡①✐st❡ ✉♠❛s❡q✉ê♥❝✐❛ ❞❡ ❝♦❜❡rt✉r❛s ❛❜❡rt❛s V0,V1,V2, ....,Vn, ... ❞❡ X t❛❧ q✉❡ V0 = U

❡ Vn+1 é r❡✜♥❛♠❡♥t♦ ❡str❡❧❛ ❞❡ Vn✳

❖ ♣ró①✐♠♦ r❡s✉❧t❛❞♦ ♥♦s ❣❛r❛♥t❡ q✉❡ ❜❛✐①♦ ❝❡rt❛s ❝♦♥❞✐çõ❡s✱ é ♣♦ssí✈❡❧❡♥❝♦♥tr❛r ✉♠ r❡✜♥❛♠❡♥t♦ ❛❜❡rt♦ ❧♦❝❛❧♠❡♥t❡ ✜♥✐t♦✳

Pr♦♣♦s✐çã♦ ✷✳✷✹✳ ❙❡ (X,D) é ✉♠ ❡s♣❛ç♦ ✉♥✐❢♦r♠❡ ❡ D ∈ D é ✉♠ ❛❜❡rt♦✱ ❡♥tã♦❛ ❢❛♠í❧✐❛ V = {D[x] : x ∈ X} ♣♦ss✉✐ ✉♠ r❡✜♥❛♠❡♥t♦ ❛❜❡rt♦ ❧♦❝❛❧♠❡♥t❡ ✜♥✐t♦ U✳

❉❡♠♦♥str❛çã♦✳ ❙✉♣♦♥❤❛ D ✉♠ ❛❜❡rt♦ ❡♠ X ×X✳ P❛r❛ t♦❞♦ x ∈ X t❡♠✲s❡ q✉❡D[x] é ✉♠ ❝♦♥❥✉♥t♦ ❛❜❡rt♦ ❡♠ X✳ ❆❣♦r❛✱ t♦♠❡ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❞✐❛❣♦♥❛❧ ❛❜❡rt❛s✐♠étr✐❝❛ E ❡♠ D t❛❧ q✉❡ 4E ⊆ D✳ ❊♥tã♦

E = {E[x] : x ∈ X}

é ✉♠ r❡✜♥❛♠❡♥t♦ ❡str❡❧❛ ❞❡ V = {D[x] : x ∈ X} ❡ V é ✉♠❛ ❝♦❜❡rt✉r❛ ♥♦r♠❛❧ ♣❛r❛X✳ ■st♦ ✐♠♣❧✐❝❛ q✉❡✱ V ♣♦ss✉✐ ✉♠ r❡✜♥❛♠❡♥t♦ ❛❜❡rt♦ ❧♦❝❛❧♠❡♥t❡ ✜♥✐t♦ U✳

✷✶

❈❛♣ít✉❧♦ ✸

❏♦❣♦ ❚♦♣♦❧ó❣✐❝♦ Pr♦①✐♠❛❧

◆❡st❛ ❝❛♣ít✉❧♦✱ ❞❡✜♥✐r❡♠♦s ♦ ❥♦❣♦ ♣r♦①✐♠❛❧ ❡ t❡♥❞♦ ❞❡✜♥✐❞♦ ♣❛ss❛r❡♠♦s ❛❞❡✜♥✐r ✉♠ ♥♦✈♦ ❡s♣❛ç♦ ❝❤❛♠❛❞♦ ❡s♣❛ç♦ ♣r♦①✐♠❛❧✱ ❛❧é♠ ❞✐ss♦ ✈❡r❡♠♦s ❛❧❣✉♠❛s❝❛r❛❝t❡r✐③❛çõ❡s ❞❡st❡ ❡s♣❛ç♦✳ ❆ ♣r✐♥❝✐♣❛❧ r❡❢❡rê♥❝✐❛ é ❬✷❪

✸✳✶ ❏♦❣♦ Pr♦①✐♠❛❧

◆❡st❛ s❡❝çã♦✱ ❞❡✜♥✐r❡♠♦s ♦ ❥♦❣♦ t♦♣♦❧ó❣✐❝♦ ✐♥✜♥✐t♦ ✭❥♦❣♦ ♣r♦①✐♠❛❧✮ ❡♥tr❡ ❞♦✐s❥♦❣❛❞♦r❡s A ❡ B ♥✉♠ ❡s♣❛ç♦ ✉♥✐❢♦r♠❡ ❡ s✉❛s r❡❣r❛s✳ ❆♥t❡s ❞❡ ❞❡✜♥✐r ❢♦r♠❛❧♠❡♥t❡♣♦❞❡♠♦s ♦❜s❡r✈❛r ♦ ❥♦❣♦ ♣r♦①✐♠❛❧ ❣r❛✜❝❛♠❡♥t❡ ♥♦ s❡❣✉✐♥t❡ ❞❡s❡♥❤♦✳

❋✐❣✉r❛ ✸✳✶✿ ❏♦❣♦ ♣r♦①✐♠❛❧

◆♦ ❞❡s❡♥❤♦✱ ♦ ❥♦❣♦ ♣r♦①✐♠❛❧ ❡♥tr❡ ♦s ❥♦❣❛❞♦r❡s A ❡ B ♥✉♠ ❡s♣❛ç♦s ✉♥✐❢♦r♠❡sã♦ ❡s❝♦❧❤❛s ✐♥✜♥✐t❛s✱ ♦♥❞❡ ♦ ❥♦❣❛❞♦r A ❡s❝♦❧❤❡ ✈✐③✐♥❤❛♥ç❛s ❞✐❛❣♦♥❛✐s ✭❡❧❡♠❡♥t♦s❞❛ ✉♥✐❢♦r♠✐❞❛❞❡ ❡♠ X✮ ❡ ♦ ❥♦❣❛❞♦r B ❡s❝♦❧❤❡ ♦s ♣♦♥t♦s ❞❡ X✳

✷✷

✷✸ ✸✳✶✳ ❏❖●❖ P❘❖❳■▼❆▲

❉❡✜♥✐çã♦ ✸✳✶✳ ❙❡❥❛ (X,D) ✉♠ ❡s♣❛ç♦ ✉♥✐❢♦r♠❡✳ ❉❡✜♥❛✲s❡ ❥♦❣♦ ♣r♦①✐♠❛❧ ❡♥tr❡♦s ❥♦❣❛❞♦r❡s A ❡ B ❡♠ (X,D) ♦✉✱ ❥♦❣♦ ♣r♦①✐♠❛❧✱ ❝♦♠♦ ❛s ❡s❝♦❧❤❛s ✐♥✜♥✐t❛s ❞♦s❥♦❣❛❞♦r❡s A ❡ B ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿

✶✮ ❏♦❣❛❞♦r A✿ ❡s❝♦❧❤❡ X ×X❀❏♦❣❛❞♦r B✿ ❡s❝♦❧❤❡ x1 ∈ X❀

✷✮ ❏♦❣❛❞♦r A✿ ❡s❝♦❧❤❡ D1 ∈ D❀❏♦❣❛❞♦r B✿ ❡s❝♦❧❤❡ x2 ∈ X❀

✸✮ ❏♦❣❛❞♦r A✿ ❡s❝♦❧❤❡ D2 ∈ D ❝♦♠ D2 ⊆ D1❀❏♦❣❛❞♦r B✿ ❡s❝♦❧❤❡ x3 ∈ D1[x2]❀

✳✳✳

♥✮ ❏♦❣❛❞♦r A✿ ❡s❝♦❧❤❡ Dn−1 ∈ D ❝♦♠ Dn−1 ⊆ ... ⊆ D2 ⊆ D1❀❏♦❣❛❞♦r B✿ ❡s❝♦❧❤❡ xn ∈ Dn−2[xn−1]❀

✳✳✳

❆s ❥♦❣❛❞❛s ❞❡ A ❡ B ♥❡st❡ ❥♦❣♦ ♣r♦①✐♠❛❧ é ✐♥✜♥✐t❛✱ ♦♥❞❡ ♦ ❝♦♥❥✉♥t♦ ❞❡ ❥♦❣❛❞❛s❡s❝♦❧❤✐❞❛s ♣❡❧♦s ❥♦❣❛❞♦r❡s sã♦ ❝❤❛♠❛❞❛s ❞❡ ❡str❛té❣✐❛s✳

❯♠❛ ✈❡③ t❡♥❞♦ ❞❡✜♥✐❞♦ ❢♦r♠❛❧♠❡♥t❡ ♦ ❥♦❣♦ ♣r♦①✐♠❛❧ ❡♥tr❡ ♦s ❥♦❣❛❞♦r❡s A❡ B ♥✉♠ ❡s♣❛ç♦ ✉♥✐❢♦r♠❡ ♦✉ s✐♠♣❧❡s♠❡♥t❡ ❥♦❣♦ ♣r♦①✐♠❛❧✱ ❞❡✜♥✐r❡♠♦s s♦❜r❡q✉❛✐s ❝♦♥❞✐çõ❡s ♦ ❥♦❣❛❞♦r A ✈❡♥❝❡ ♦ ❥♦❣♦ ♣r♦①✐♠❛❧✳ ◆❡st❡ tr❛❜❛❧❤♦ ♥ã♦ ❡st❛♠♦s✐♥t❡r❡ss❛❞♦s s❡ ♦ ❥♦❣❛❞♦r B ✈❡♥❝❡ ♦ ❥♦❣♦ ♣r♦①✐♠❛❧✳

❉❡✜♥✐çã♦ ✸✳✷✳ ❙❡❥❛ X ✉♠ ❡s♣❛ç♦ ✉♥✐❢♦r♠❡✳ ❖ ❥♦❣❛❞♦r A ✈❡♥❝❡ ♦ ❥♦❣♦ ♣r♦①✐♠❛❧s❡✱ ❛s ❡s❝♦❧❤❛s ✐♥✜♥✐t❛s ❞♦s ❥♦❣❛❞♦r❡s A ❡ B s❛t✐s❢❛③ ❛ s❡❣✉✐♥t❡ ♣r♦♣r✐❡❞❛❞❡✿

✶✮ ❆ s❡q✉ê♥❝✐❛ (x1, x2, ..., xn, ...) ❝♦♥✈❡r❣❡ ❡♠ X✱ ♦✉

✷✮⋂

n∈N

Dn−1[xn] = ∅✳

◆❛ s❡❣✉✐♥t❡ ❞❡✜♥✐çã♦ ✜①❛r❡♠♦s ❛❧❣✉♠❛s ❞❡✜♥✐çõ❡s s♦❜r❡ ❡❧❡♠❡♥t♦s ❞♦ ❥♦❣♦♣r♦①✐♠❛❧✳

❉❡✜♥✐çã♦ ✸✳✸✳ ❙❡❥❛ (X,D) ✉♠ ❡s♣❛ç♦ ✉♥✐❢♦r♠❡✳ ❈♦♥s✐❞❡r❡ ♦ ❥♦❣♦ ♣r♦①✐♠❛❧ ❞❛❉❡✜♥✐çã♦ ✸✳✶✳ ❊♥tã♦✿

✶✮ ❆ s❡q✉ê♥❝✐❛ ✐♥✜♥✐t❛ (x1, x2, ..., xn, ..) ❢♦r♠❛❞❛ ♣❡❧❛s ❡s❝♦❧❤❛s ❞♦ ❥♦❣❛❞♦r Bé ❝❤❛♠❛❞❛ ❞❡ s❡q✉ê♥❝✐❛ ♣r♦①✐♠❛❧✳

✷✮ ❆ s❡q✉ê♥❝✐❛ ✜♥✐t❛ (x1, D1, x2, D2, ..., Dn− 1, xn) ❢♦r♠❛❞❛ ♣❡❧❛s ❡s❝♦❧❤❛s ❞♦s❥♦❣❛❞♦r❡s A ❡ B é ❝❤❛♠❛❞❛ ❞❡ ❥♦❣♦ ♣❛r❝✐❛❧ ♣r♦①✐♠❛❧ ❞♦ ❥♦❣♦ ♣r♦①✐♠❛❧✳

✸✮ ❆ s❡q✉ê♥❝✐❛ ✜♥✐t❛ (x1, x2, ..., xn) ❢♦r♠❛❞❛ ♣❡❧❛s ❡s❝♦❧❤❛s ❞♦ ❥♦❣❛❞♦r B é❝❤❛♠❛❞❛ ❞❡ s❡q✉ê♥❝✐❛ ❛❞♠✐ssí✈❡❧✳

✷✸

✷✹ ✸✳✶✳ ❏❖●❖ P❘❖❳■▼❆▲

❉❡✜♥✐çã♦ ✸✳✹✳ ❖ ❝♦♥❥✉♥t♦ ❞❡ t♦❞❛s ❛s s❡q✉ê♥❝✐❛s ❛❞♠✐ssí✈❡✐s ♥♦ ❥♦❣♦ ♣r♦①✐♠❛❧é ❝❤❛♠❛❞♦ ❞❡ ❝♦♥❥✉♥t♦ ❛❞♠✐ssí✈❡❧ ❡ s❡rá ❞❡♥♦t❛❞♦ ♣♦r A✳

❈♦♠♦ ❡♠ t♦❞♦ ❥♦❣♦ ❡①✐st❡♠ ❛s ❡str❛té❣✐❛s✱ ♥❛ ❞❡✜♥✐çã♦ ❞❡ ❥♦❣♦ ♣r♦①✐♠❛❧ ✈✐♠♦sq✉❡ ♦ ❝♦♥❥✉♥t♦ ❞❡ ❡s❝♦❧❤❛s ❞❡ ❝❛❞❛ ❥♦❣❛❞♦r é ✉♠❛ ❡str❛té❣✐❛✳ ❆❣♦r❛ ❢♦r♠❛❧♠❡♥t❡❞❡✜♥✐r❡♠♦s ❛ ❡str❛té❣✐❛ ♣❛r❛ ♦ ❥♦❣❛❞♦r A ♥♦ ❥♦❣♦ ♣r♦①✐♠❛❧✱ ♣♦r ♠❡✐♦ ❞❛s ❡s❝♦❧❤❛s❞♦ ❥♦❣❛❞♦r A✱ ♦ q✉❛❧ s❡rá ❞❡s❝r✐t❛ ♣❡❧❛ ❢✉♥çã♦ ❛ s❡❣✉✐r✿

❉❡✜♥✐çã♦ ✸✳✺✳ ❙❡❥❛ (X,D) ✉♠ ❡s♣❛ç♦ ✉♥✐❢♦r♠❡✳ ❉❡✜♥❛✲s❡ ❛ ❡str❛té❣✐❛ ♣❛r❛ ♦❥♦❣❛❞♦r A ♥♦ ❥♦❣♦ ♣r♦①✐♠❛❧ ❝♦♠♦ ✉♠❛ ❢✉♥çã♦✿

ω : A −→ D

t❛❧ q✉❡✿

✶✮ ω(∅) = X ×X✳

✷✮ xn+1 ∈ ω(x1, x2, ..., xn−1)[xn]✳

✸✮ ω(x1, x2, ..., xn) ⊆ ω(x1, x2, ..., xn−1)✱

♦♥❞❡ A é ♦ ❝♦♥❥✉♥t♦ ❛❞♠✐ssí✈❡❧ ♥♦ ❥♦❣♦ ♣r♦①✐♠❛❧ ❡♠ X✳

❇❛s✐❝❛♠❡♥t❡✱ ♣♦❞❡♠♦s ♦❜s❡r✈❛r q✉❡ ❛s ❝♦♥❞✐çõ❡s ❞♦ ❥♦❣♦ sã♦ ❛s ❡str❛té❣✐❛s❞♦ ❥♦❣❛❞♦r A✳ P❛r❛ ✜♥❛❧✐③❛r ♣❛ss❛♠♦s ❛ ❞❡✜♥✐r ✉♠❛ ❡str❛té❣✐❛ ✈❡♥❝❡❞♦r❛ ♣❛r❛ ♦❥♦❣❛❞♦r A ♥♦ ❥♦❣♦ ♣r♦①✐♠❛❧✳ ❆♥t❡s ❞❡ ♣❛ss❛r ♥❛ s❡❣✉✐♥t❡ ❞❡✜♥✐çã♦✱ ❛ ♣♦st❡r✐♦r✐✈❡r❡♠♦s ❛s ❝♦♥❞✐çõ❡s ❞❡ ✈❡♥❝❡r ♦ ❥♦❣♦ ♣r♦①✐♠❛❧✳

❉❡✜♥✐çã♦ ✸✳✻✳ ❙❡❥❛♠ (X,D) ✉♠ ❡s♣❛ç♦ ✉♥✐❢♦r♠❡ ❡ ω : A −→ D ✉♠❛ ❡str❛té❣✐❛♥♦ ❥♦❣♦ ♣r♦①✐♠❛❧✱ ♦♥❞❡ A é ♦ ❝♦♥❥✉♥t♦ ❛❞♠✐ssí✈❡❧✳ ❊♥tã♦✿

✶✮ ω é ❡str❛té❣✐❛ ✈❡♥❝❡❞♦r❛ ♦✉ D✲✈❡♥❝❡❞♦r❛✱ s❡ t♦❞❛ s❡q✉ê♥❝✐❛ ♣r♦①✐♠❛❧(x1, x2, ..., xn, ...) ♦✉ ❝♦♥✈❡r❣❡ ❡♠ X ♦✉

⋂

n∈N

ω(x1, x2, ..., xn−1)[xn] = ∅✳

✷✮ ω é ✉♠❛ ❡str❛té❣✐❛ ❛❜s♦❧✉t❛♠❡♥t❡ ✈❡♥❝❡❞♦r❛ ♦✉ ❛❜s♦❧✉t❛♠❡♥t❡ D✲✈❡♥❝❡❞♦r❛✱ s❡ t♦❞❛ s❡q✉ê♥❝✐❛ ♣r♦①✐♠❛❧ (x1, x2, ..., xn, ...) ❝♦♥✈❡r❣❡ ❡♠ X✳

✸✮ ω é ✉♠❛ ❡str❛té❣✐❛ q✉❛s❡ ✈❡♥❝❡❞♦r❛ ♦✉ q✉❛s❡ D✲✈❡♥❝❡❞♦r❛✱ s❡ t♦❞❛s❡q✉ê♥❝✐❛ ♣r♦①✐♠❛❧ (x1, x2, ..., xn, ...)✱ ♦✉ t❡♠ ✉♠ ♣♦♥t♦ ❞❡ ❛❝✉♠✉❧❛çã♦ ❡♠X ♦✉

⋂

n∈N

ω(x1, x2, ..., xn−1)[xn] = ∅✳

❱❡r❡♠♦s q✉❡ ❛s ❝♦♥❞✐çõ❡s ❞❡ ✈❡♥❝❡r ✉♠ ❥♦❣♦ ✭❡♠ ❣❡r❛❧✮ ❞❡✜♥❡♠ ♥♦✈♦s ❡s♣❛ç♦st♦♣♦❧ó❣✐❝♦s ✭❡s♣❛ç♦ ♣r♦①✐♠❛❧✱ q✉❛s❡ ♣r♦①✐♠❛❧✱ ❛❜s♦❧✉t❛♠❡♥t❡ ♣r♦①✐♠❛❧✮✳ P♦r❡♠♣❛r❛ s❡✉ ♠❡❧❤♦r ❡♥t❡♥❞✐♠❡♥t♦✱ ✈❡r❡♠♦s ✉♠ ❡①❡♠♣❧♦ ❜ás✐❝♦ ❞❡ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦♣r♦①✐♠❛❧✳

✷✹

✷✺ ✸✳✷✳ ❊❙P❆➬❖❙ P❘❖❳■▼❆▲

❊①❡♠♣❧♦ ✸✳✼✳ ❙❡❥❛ X ✉♠ s✉❜❝♦♥❥✉♥t♦ ♥ã♦ ✈❛③✐♦ ❞❡ R ❡ Dd ✉♠❛ ✉♥✐❢♦r♠✐❞❛❞❡✐♥❞✉③✐❞❛ ♣❡❧❛ ♠étr✐❝❛ d ❡♠ X✱ ❝♦♠ ❡❧❡♠❡♥t♦s ❞❛ ❢♦r♠❛✿

Dn = {(x, y) ∈ X ×X : d(x, y) < 2−n}.

❈♦♥s✐❞❡r❡ ♦ ❥♦❣♦ ♣r♦①✐♠❛❧ ❡♥tr❡ ♦s ❥♦❣❛❞♦r❡s A ❡ B ❝♦♠♦ ❛s ❡s❝♦❧❤❛s ✐♥✜♥✐t❛s ❞❛s❡❣✉✐♥t❡ ❢♦r♠❛✿

✶✮ ❏♦❣❛❞♦r A✿ ❡s❝♦❧❤❡ X ×X❀❏♦❣❛❞♦r B✿ ❡s❝♦❧❤❡ x1 ∈ X❀

✷✮ ❏♦❣❛❞♦r A✿ ❡s❝♦❧❤❡ D1 ∈ D❀❏♦❣❛❞♦r B✿ ❡s❝♦❧❤❡ x2 ∈ X❀

✸✮ ❏♦❣❛❞♦r A✿ ❡s❝♦❧❤❡ D2 ∈ D ❝♦♠ D2 ⊆ D1❀❏♦❣❛❞♦r B✿ ❡s❝♦❧❤❡ x3 ∈ D1[x2]❀

✳✳✳

♥✮ ❏♦❣❛❞♦r A✿ ❡s❝♦❧❤❡ Dn−1 ∈ D ❝♦♠ Dn−1 ⊆ ... ⊆ D2 ⊆ D1❀❏♦❣❛❞♦r B✿ ❡s❝♦❧❤❡ xn ∈ Dn−2[xn−1]❀

✳✳✳

❆ss✐♠ ♣❛r❛ q✉❛❧q✉❡r ❡s❝♦❧❤❛ ❞♦ ❥♦❣❛❞♦r B ❝♦♠ ❛s r❡❣r❛s ❛❝✐♠❛ t❡♠✲s❡ q✉❡ t♦❞❛s❡q✉ê♥❝✐❛ ♣r♦①✐♠❛❧ (x1, x2, ..., xn, ...) ❝♦♥✈❡r❣❡ ❛ ❛❧❣✉♠ ♣♦♥t♦ ❡♠ X✳ P♦rt❛♥t♦✱ ♦❥♦❣❛❞♦r A ♣♦ss✉✐ ✉♠❛ ❡str❛té❣✐❛ ❛❜s♦❧✉t❛♠❡♥t❡ ✈❡♥❝❡❞♦r❛✳

✸✳✷ ❊s♣❛ç♦s Pr♦①✐♠❛❧

◆❡st❛ s❡çã♦✱ ✉♠❛ ✈❡③ ❞❡✜♥✐❞♦ ✉♠ ❥♦❣♦ t♦♣♦❧ó❣✐❝♦ ✐♥✜♥✐t♦✱ ♣♦❞❡♠♦s ❞❡✜♥✐r♥♦✈♦s ❡s♣❛ç♦s ✭❡s♣❛ç♦s D✲♣r♦①✐♠❛❧✱ ❛❜s♦❧✉t❛♠❡♥t❡ D✲♣r♦①✐♠❛❧ ❡ q✉❛s❡ D✲♣r♦①✐♠❛❧✮ ❡♠ ❢✉♥çã♦ ❞❛ s✉❛ ✉♥✐❢♦r♠✐❞❛❞❡✱ ❡ ♣♦st❡r✐♦r♠❡♥t❡ ❞❡✜♥✐r❡♠♦s ❡st❡s❡s♣❛ç♦s ❡♠ ❢✉♥çã♦ ❞❛ t♦♣♦❧♦❣✐❛ ❣❡r❛❞❛ ♣❡❧❛ ✉♥✐❢♦r♠✐❞❛❞❡ ✭❡s♣❛ç♦s ♣r♦①✐♠❛❧✱❛❜s♦❧✉t❛♠❡♥t❡ ♣r♦①✐♠❛❧ ❡ q✉❛s❡ ♣r♦①✐♠❛❧✮✳ ❚❛♠❜é♠ ✈❡r❡♠♦s ❛❧❣✉♥s ❡①❡♠♣❧♦s❜ás✐❝♦s ❞❡st❡s ❡s♣❛ç♦s✳ ❬✷❪

❉❡✜♥✐çã♦ ✸✳✽✳ ❙❡❥❛♠ X ✉♠ ❡s♣❛ç♦ ✉♥✐❢♦r♠❡ ❡ ω : A −→ D ✉♠❛ ❡str❛té❣✐❛ ♣❛r❛♦ ❥♦❣❛❞♦r A ♥♦ ❥♦❣♦ ♣r♦①✐♠❛❧✳ ❊♥tã♦✿

✶✮ X é ✉♠ ❡s♣❛ç♦ D✲♣r♦①✐♠❛❧✱ s❡ ❡①✐st❡ ✉♠❛ ❡str❛té❣✐❛ D✲✈❡♥❝❡❞♦r❛ ω ♣❛r❛♦ ❥♦❣❛❞♦r A ♥♦ ❥♦❣♦ ♣r♦①✐♠❛❧✳

✷✮ X é ✉♠ ❡s♣❛ç♦ ❛❜s♦❧✉t❛♠❡♥t❡ D✲♣r♦①✐♠❛❧✱ s❡ ❡①✐st❡ ✉♠❛ ❡str❛té❣✐❛❛❜s♦❧✉t❛♠❡♥t❡ D✲✈❡♥❝❡❞♦r❛ ω ♣❛r❛ ♦ ❥♦❣❛❞♦r A ♥♦ ❥♦❣♦ ♣r♦①✐♠❛❧✳

✸✮ X é ✉♠ ❡s♣❛ç♦ q✉❛s❡ D✲♣r♦①✐♠❛❧✱ s❡ ❡①✐st❡ ✉♠❛ ❡str❛té❣✐❛ q✉❛s❡ D✲✈❡♥❝❡❞♦r❛ ω ♣❛r❛ ♦ ❥♦❣❛❞♦r A ♥♦ ❥♦❣♦ ♣r♦①✐♠❛❧✳

✷✺

✷✻ ✸✳✷✳ ❊❙P❆➬❖❙ P❘❖❳■▼❆▲

❈♦♠♦ t♦❞❛ ✉♥✐❢♦r♠✐❞❛❞❡ ✐♥❞✉③ ✉♠❛ t♦♣♦❧♦❣✐❛✱ ❡♥tã♦ ♣❛ss❛♠♦s ❛ ❞❡✜♥✐r ♦ss❡❣✉✐♥t❡s ❡s♣❛ç♦s ❡♠ ❢✉♥çã♦ ❞❛ t♦♣♦❧♦❣✐❛ ✐♥❞✉③✐❞❛ ♣❡❧❛ ✉♥✐❢♦r♠✐❞❛❞❡✳

❉❡✜♥✐çã♦ ✸✳✾✳ ❙❡❥❛♠ (X,D) ✉♠ ❡s♣❛ç♦ ✉♥✐❢♦r♠❡ ❡ ω : A −→ D ✉♠❛ ❡str❛té❣✐❛♣❛r❛ ♦ ❥♦❣❛❞♦r A ♥♦ ❥♦❣♦ ♣r♦①✐♠❛❧✱ ♦♥❞❡ A é ♦ ❝♦♥❥✉♥t♦ ❛❞♠✐ssí✈❡❧✳

✶✮ ❯♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ X é ✉♠ ❡s♣❛ç♦ ♣r♦①✐♠❛❧✱ s❡ ❡①✐st❡ ✉♠❛✉♥✐❢♦r♠✐❞❛❞❡ D ❡♠ X q✉❡ ✐♥❞✉③ ❛ t♦♣♦❧♦❣✐❛ ❡♠ X t❛❧ q✉❡ X ✉♠ ❡s♣❛ç♦D✲♣r♦①✐♠❛❧✳

✷✮ ❯♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ X é ✉♠ ❡s♣❛ç♦ ❛❜s♦❧✉t❛♠❡♥t❡ ♣r♦①✐♠❛❧✱ s❡ ❡①✐st❡✉♠❛ ✉♥✐❢♦r♠✐❞❛❞❡ D ❡♠ X q✉❡ ✐♥❞✉③ ❛ t♦♣♦❧♦❣✐❛ ❡♠ X t❛❧ q✉❡ X ✉♠ ❡s♣❛ç♦❛❜s♦❧✉t❛♠❡♥t❡ D✲♣r♦①✐♠❛❧✳

✸✮ ❯♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ X é ✉♠ ❡s♣❛ç♦ q✉❛s❡ ♣r♦①✐♠❛❧✱ s❡ ❡①✐st❡ ✉♠❛✉♥✐❢♦r♠✐❞❛❞❡ D ❡♠ X q✉❡ ✐♥❞✉③ ❛ t♦♣♦❧♦❣✐❛ ❡♠ X t❛❧ q✉❡ X ✉♠ ❡s♣❛ç♦q✉❛s❡ D✲♣r♦①✐♠❛❧✳

❈♦♠♦ ♣❛r❛ t♦❞❛ ✉♥✐❢♦r♠✐❞❛❞❡ ❡①✐st❡ ✉♠❛ ❜❛s❡ ✉♥✐❢♦r♠❡✳ P♦❞❡♠♦s ❛✜r♠❛r ♦ss❡❣✉✐♥t❡s r❡s✉❧t❛❞♦s✳

Pr♦♣♦s✐çã♦ ✸✳✶✵✳ ❙❡ X é ✉♠ ❡s♣❛ç♦ D✲♣r♦①✐♠❛❧ ❡ E é ✉♠❛ ❜❛s❡ ♣❛r❛ ❛✉♥✐❢♦r♠✐❞❛❞❡ D✱ ❡♥tã♦ X é ✉♠ ❡s♣❛ç♦ E✲♣r♦①✐♠❛❧✳

❉❡♠♦♥str❛çã♦✳ ❙✉♣♦♥❤❛ q✉❡ X é ✉♠ ❡s♣❛ç♦ D✲♣r♦①✐♠❛❧✱ ❡♥tã♦ ❡①✐st❡ ✉♠❛❡str❛té❣✐❛ D✲✈❡♥❝❡❞♦r❛ ω : A −→ D ♥♦ ❥♦❣♦ ♣r♦①✐♠❛❧✳❙❡❥❛ E ✉♠❛ ❜❛s❡ ♣❛r❛ ❛ ✉♥✐❢♦r♠✐❞❛❞❡ D✳ ❉❡✜♥❛✲s❡ ❛ ❡str❛té❣✐❛ E✲✈❡♥❝❡❞♦r❛✿

υ : A −→ E

❝♦♠✱ υ(x1, x2, ..., xn) ⊆ ω(x1, x2..., xn)✱ t❛❧ q✉❡✿

✶✮ υ(∅) = ω(∅) = X ×X✳

✷✮ ❙❡ (x1, x2, ..., xn) ✉♠❛ s❡q✉ê♥❝✐❛ ❛❞♠✐ssí✈❡❧✱ ❝♦♠ xn+1 ∈ω(x1, x2, ..., xn−1)[xn]✱ ❡♥tã♦ ❡①✐st❡ ✉♠ ❡❧❡♠❡♥t♦ υ(x1, x2, ..., xn−1)[xn] ❞❛❜❛s❡ E t❛❧ q✉❡ xn+1 ∈ υ(x1, x2, ..., xn−1)[xn] ⊆ ω(x1, x2, ..., xn−1)[xn]✳

✸✮ P❛r❛ t♦❞♦ ω(x1, x2, ..., xn) ⊆ ω(x1, x2..., xn−1)✱ ❡①✐st❡♠ υ(x1, x2, ..., xn) ⊆ω(x1, x2, ..., xn) ❡ υ(x1, x2, ..., xn−1) ⊆ ω(x1, x2, ..., xn−1)✱ ♣❡❧❛ ❉❡✜♥✐çã♦ ✸✳✶t❡♠✲s❡ q✉❡✱ υ(x1, x2, ..., xn) ⊆ υ(x1, x2, ..., xn−1)✳

✹✮ ❙❡ (x1, x2, ..., xn, ...) ✉♠ s❡q✉ê♥❝✐❛ ♣r♦①✐♠❛❧ ❡ ❞❡s❞❡ q✉❡ ♣❛r❛ t♦❞♦n ∈ N t❡♠✲s❡ q✉❡ υ(x1, x2, ..., xn−1)[xn] ⊆ ω(x1, x2..., xn−1)[xn]✱⋂

n∈N

ω(x1, x2, ..., xn−1)[xn] = ∅✱ ♣♦✐s ω é ✉♠❛ ❡str❛té❣✐❛ D✲✈❡♥❝❡❞♦r❛ ♥♦ ❥♦❣♦

♣r♦①✐♠❛❧✱ ❡♥tã♦⋂

n∈N

υ(x1, x2, ..., xn−1)[xn] = ∅✳

P♦rt❛♥t♦✱ υ é ✉♠❛ ❡str❛té❣✐❛ E✲✈❡♥❝❡❞♦r❛ ❡ ❛ss✐♠ X t❛♠❜é♠ é ✉♠ ❡s♣❛ç♦ E✲♣r♦①✐♠❛❧✳

✷✻

✷✼ ✸✳✷✳ ❊❙P❆➬❖❙ P❘❖❳■▼❆▲

◆❛ s❡❣✉✐♥t❡ ♣r♦♣♦s✐çã♦ ♠♦str❛r❡♠♦s q✉❡ ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦ X é ✉♠ ❡s♣❛ç♦Dd✲♣r♦①✐♠❛❧ ❝♦♠ ❛ ✉♥✐❢♦r♠✐❞❛❞❡ ✐♥❞✉③✐❞❛ ♣❡❧❛ s✉❛ ♠étr✐❝❛✱ q✉❡ ✐♥❞✉③ ✉♠❛✉♥✐❢♦r♠✐❞❛❞❡ s♦❜r❡ ♦ ❡s♣❛ç♦✳

Pr♦♣♦s✐çã♦ ✸✳✶✶✳ ❙❡ X é ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦ ❡ Dd é ❛ ✉♥✐❢♦r♠✐❞❛❞❡ ✐♥❞✉③✐❞❛♣❡❧❛ ♠étr✐❝❛ d ❡♠ X✱ ❡♥tã♦ X é ✉♠ ❡s♣❛ç♦ D✲♣r♦①✐♠❛❧✳

❉❡♠♦♥str❛çã♦✳ ❙❡❥❛♠ X ✉♠ ❡s♣❛ç♦ ✉♥✐❢♦r♠❡ ❝♦♠ ❛ ✉♥✐❢♦r♠✐❞❛❞❡ Dd ✐♥❞✉③✐❞❛♣❡❧❛ ♠étr✐❝❛ d ❡♠ X✳ ❖❜s❡r✈❡ q✉❡ ❞❛ Pr♦♣♦s✐çã♦ ✷✳✶✸ ❝✉❥♦s ❡❧❡♠❡♥t♦s ❞❡ Dd sã♦❞❛ ❢♦r♠❛✿

Dn+1 = {(x, y) ∈ X ×X : d(x, y) < 2−(n+1)}.

❉❡✜♥❛✲s❡ ❛ ❡str❛té❣✐❛ ω : A −→ Dd ❞❛❞❛ ♣♦r ω(x1, ..., xn) = Dn+1 t❛❧ q✉❡✿

✶✮ ω(∅) = X ×X✳

✷✮ P❡❧❛ ❞❡✜♥✐çã♦ ❞❡ ❥♦❣♦ ♣r♦①✐♠❛❧✱ t❡♠✲s❡ xn+1 ∈ ω(x1, x2, ..., xn−1)[xn]✳

✸✮ ❙❡ ω(x1, ..., xn−1) = Dn ❡ ω(x1, ..., xn) = Dn+1 ❡ ♦❜s❡r✈❡ q✉❡ Dn+1 ⊆ Dn

♣♦✐s

(x, y) ∈ Dn+1 =⇒ d(x, y) < 2−(n+1) < 2−n

=⇒ (x, y) ∈ Dn.

❆ss✐♠ ω(x1, ..., xn) ⊆ ω(x1, ..., xn−1)✳

✹✮ ❙❡❥❛ (x1, x2, ..., xn, ...) ✉♠❛ s❡q✉ê♥❝✐❛ ♣r♦①✐♠❛❧ ❡ s✉♣♦♥❤❛✲s❡ q✉❡⋂

n∈N

ω(x1, x2, ..., xn−1)[xn] =⋂

n∈N

Dn[xn] 6= ∅ ❡♥tã♦ ♣❛r❛ t♦❞♦ n ∈ N ❡①✐st❡✳

z ∈ Dn[xn] =⇒ d(xn, z) < 2−n

=⇒ d(xn, z) −→ 0, q✉❛♥❞♦ n −→ ∞.

❧♦❣♦✱ ❛ s❡q✉ê♥❝✐❛ ♣r♦①✐♠❛❧ (x1, x2, ..., xn, ...) ❝♦♥✈❡r❣❡ ❛ z✳

P♦rt❛♥t♦✱ ω é ✉♠❛ ❡str❛té❣✐❛ Dd✲✈❡♥❝❡❞♦r❛✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡ X é ✉♠ ❡s♣❛ç♦Dd✲♣r♦①✐♠❛❧✳

◆♦ ❡♥t❛♥t♦✱ é ♦ ❝❛s♦ ❞❡ q✉❡ ❡①✐st❡ ✉♠❛ ❡str❛té❣✐❛ D✲✈❡♥❝❡❞♦r❛ ♥♦ ❥♦❣♦♣r♦①✐♠❛❧✱ ♦♥❞❡ X é ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦ ❡ Dd é ✉♠❛ ✉♥✐❢♦r♠✐❞❛❞❡ ❝♦♠♣❛tí✈❡❧❝♦♠ ❛ ✉♥✐❢♦r♠✐❞❛❞❡ ✐♥❞✉③✐❞❛ ♣❡❧❛ ♠étr✐❝❛ d ❞♦ ❡s♣❛ç♦ X✳

❆❣♦r❛ ✈❡r❡♠♦s ✉♠ ❡①❡♠♣❧♦ ♦♥❞❡ ❡①✐st❡♠ ❡s♣❛ç♦s ♠étr✐❝♦s q✉❡ ♥ã♦ ♣♦ss✉❡♠❡str❛té❣✐❛ ✈❡♥❝❡❞♦r❛ ❡♠ r❡❧❛çã♦ ❛ ❛❧❣✉♠❛ ✉♥✐❢♦r♠✐❞❛❞❡ ❞✐❢❡r❡♥t❡ ❞❛ ✉♥✐❢♦r♠✐❞❛❞❡♥❛t✉r❛❧ ✭✐♥❞✉③✐❞❛ ♣❡❧❛ ❞♦ ❡s♣❛ç♦✮✳

❊①❡♠♣❧♦ ✸✳✶✷✳ ❙❡❥❛ W ✉♠ ❝♦♥❥✉♥t♦ ♥ã♦ ❡♥✉♠❡rá✈❡❧✱ ♠✉♥✐❞♦ ❞❛ t♦♣♦❧♦❣✐❛❞✐s❝r❡t❛✳ ❖❜s❡r✈❛r q✉❡ W ♠✉♥✐❞♦ ❞❛ ♠étr✐❝❛ ❞✐s❝r❡t❛ é ✉♠ ❡s♣❛ç♦ ♠❡tr✐③❛✈❡❧✳❈♦♥s✐❞❡r❡♠♦s ❛ s❡❣✉✐♥t❡ ❜❛s❡ ✉♥✐❢♦r♠❡ D ❞♦ ❊①❡♠♣❧♦ ✷✳✶✶✱ ❤❡r❞❛❞❛ ♣♦r ✉♠

✷✼

✷✽ ✸✳✷✳ ❊❙P❆➬❖❙ P❘❖❳■▼❆▲

♣♦♥t♦ ❞❡ ❝♦♠♣❛❝✐✜❝❛çã♦ ❞❡ W ✱✭❯♠ ❡s♣❛ç♦ ❋♦rt é ❛ ❝♦♠♣❛❝✐✜❝❛çã♦ ❞❡ ✉♠ ♣♦♥t♦❞❡ ✉♠ ❡s♣❛ç♦ ❞✐s❝r❡t♦✮ ❢♦r♠❛❞♦ ♣❡❧♦s ❡❧❡♠❡♥t♦ ❞❛ ❢♦r♠❛✿

DF = F × F ∪ (W − F )× (W − F ),

♦♥❞❡ F ❡ ✉♠ s✉❜❝♦♥❥✉♥t♦ ✜♥✐t♦ ❞❡ W ✳ ❈♦♠♦ W ❡stá ♠✉♥✐❞♦ ❞❡ ✉♠❛✉♥✐❢♦r♠✐❞❛❞❡✱ ♣♦❞❡♠♦s ❝♦♥s✐❞❡r❛r ♦ ❥♦❣♦ ♣r♦①✐♠❛❧ ❡♥tr❡ ♦s ❥♦❣❛❞♦r❡s A ❡ B❝♦♠ ❛s s❡❣✉✐♥t❡s ❡s❝♦❧❤❛s✿

✶✮ ❏♦❣❛❞♦r A✿ ❡s❝♦❧❤❡ DF1= F1 × F1 ∪ (W − F1)× (W − F1)

❏♦❣❛❞♦r B✿ ❡s❝♦❧❤❡ x1 ∈ W − F1

✷✮ ❏♦❣❛❞♦r A✿ ❡s❝♦❧❤❡ DF2= F2 × F2 ∪ (W − F2)× (W − F2)

❏♦❣❛❞♦r B✿ ❡s❝♦❧❤❡ x2 ∈ W − (F1 ∪ F2) t❛❧ q✉❡ x2 ∈ DF1[x1] = W − F1

✸✮ ❏♦❣❛❞♦r A✿ ❡s❝♦❧❤❡ DF3= F3 × F3 ∪ (W − F3)× (W − F3)

❏♦❣❛❞♦r B✿ ❡s❝♦❧❤❡ x3 ∈ X − (F1 ∪F2 ∪F3) t❛❧ q✉❡ x3 ∈ DF2[x2] = W −F2

✳✳✳

♥✮ ❏♦❣❛❞♦r A✿ ❡s❝♦❧❤❡ DFn= Fn × Fn ∪ (W − Fn)× (W − Fn)

❏♦❣❛❞♦r B✿ ❡s❝♦❧❤❡ xn ∈ W − (F1∪F2∪ ...∪Fn) t❛❧ q✉❡ xn ∈ DFn−1[xn−1] =

W − Fn−1

✳✳✳

❊ ❛ss✐♠ ❝♦♥t✐♥✉❛ ✐♥✜♥✐t❛♠❡♥t❡ ♦ ❥♦❣♦ ♣r♦①✐♠❛❧✳ ❊♥tã♦ ❞❡✜♥❛ ❢♦r♠❛❧♠❡♥t❡ ❛❡str❛té❣✐❛ t❡♥❞♦ ❛s ❡s❝♦❧❤❛s ❞♦s ❥♦❣❛❞♦r❡s A ❡ B✱ ❝♦♠♦ ω : A −→ D ❞❛❞❛ ♣♦r✿

ω(x1, x2, ..., xn) = DFn+1= Fn+1 × Fn+1 ∪ (W − Fn+1)× (W − Fn+1)

t❛❧ q✉❡✿

✶✮ ω(∅) = DF1✳

✷✮ P❡❧❛ ❞❡✜♥✐çã♦ ❞♦ ❥♦❣♦ ♣r♦①✐♠❛❧ t❡♠✲s❡ q✉❡ xn+1 ∈ DFn[xn] =

ω(x1, x2, ..., xn−1)[xn]✳

✸✮ P❡❧❛ ❞❡✜♥✐çã♦ ❞❡ ❥♦❣♦ ♣r♦①✐♠❛❧ t❡♠✲s❡ q✉❡ ω(x1, x2, ..., xn) ⊆ω(x1, x2, ..., xn−1)✳

✹✮ ❙❡❥❛ (x1, x2, ..., xn, ...) ❛ s❡q✉ê♥❝✐❛ ♣r♦①✐♠❛❧ ♥♦ ❥♦❣♦ ♣r♦①✐♠❛❧ ❡ ♦❜s❡r✈❡✲s❡q✉❡

⋂

n∈N

ω(x1, ..., xn−1)[xn] =⋂

n∈N

DFn[xn] =

⋂

n∈N

W − Fn 6= ∅✳ ❆❧❡♠ ❞✐ss♦ ❛

s❡q✉ê♥❝✐❛ ♣r♦①✐♠❛❧ (x1, x2, ..., xn, ...) ♥ã♦ ❝♦♥✈❡r❣❡ ❛ ♥❡♥❤✉♠ ♣♦♥t♦ ❞❡ W

P♦rt❛♥t♦✱ W ♥ã♦ é ✉♠ ❡s♣❛ç♦ D✲♣r♦①✐♠❛❧✱ ♠❡s♠♦ ❛ss✐♠ ❛ t♦♣♦❧♦❣✐❛ ❞❡ W s❡♥❞♦♠❡tr✐③❛✈é❧✳

✷✽

✷✾ ✸✳✷✳ ❊❙P❆➬❖❙ P❘❖❳■▼❆▲

❊①❡♠♣❧♦ ✸✳✶✸✳ ❙❡❥❛ X = W ∪ {∞} ✉♠ ❡s♣❛ç♦ ❋♦rt ♥ã♦ ❡♥✉♠❡rá✈❡❧ ♠✉♥✐❞♦ ❞❡✉♠❛ ❜❛s❡ ✉♥✐❢♦r♠✐❞❛❞❡ D ✷✳✶✶ ❢♦r♠❛❞♦ ♣❡❧♦s ❡❧❡♠❡♥t♦s ❞❛ ❢♦r♠❛✿

DF = F × F ∪ (W − F )× (W − F ),

♦♥❞❡ F é ✉♠ s✉❜❝♦♥❥✉♥t♦ ✜♥✐t♦ ❞❡ X✳ ❊♥tã♦ t❡♥❞♦ X ♠✉♥✐❞♦ ❞❡ ✉♠❛✉♥✐❢♦r♠✐❞❛❞❡❀ ♣❛r❛ ❝♦♥str✉✐r ❛ ❡str❛té❣✐❛ ♣♦❞❡♠♦s ❝♦♠❡ç❛r ♦ ❥♦❣♦ ♣r♦①✐♠❛❧ ❡♥tr❡♦s ❥♦❣❛❞♦r❡s A ❡ B ❝♦♠ ❛s s❡❣✉✐♥t❡s ❡s❝♦❧❤❛s✿

✶✮ ❏♦❣❛❞♦r A✿ ❡s❝♦❧❤❡ X ×X❏♦❣❛❞♦r B✿ ❡s❝♦❧❤❡ x1 ∈ X ❝♦♠ x1 6= ∞

✷✮ ❏♦❣❛❞♦r A✿ ❡s❝♦❧❤❡ ω(x1) = DF1= F1 × F1 ∪ (W − F1)× (W − F1)

❏♦❣❛❞♦r B✿ ❡s❝♦❧❤❡ x2 ∈ X ×X[x1] ❝♦♠ x2 6= x1

✸✮ ❏♦❣❛❞♦r A✿ ❡s❝♦❧❤❡ ω(x1, x2) = DF2= F2 × F2 ∪ (W − F2)× (W − F2)

❏♦❣❛❞♦r B✿ ❡s❝♦❧❤❡ x3 ∈ DF1[x2] ❝♦♠ x3 6= x2

✹✮ ❏♦❣❛❞♦r A✿ ❡s❝♦❧❤❡ ω(x1, x2, x3) = DF3= F3 × F3 ∪ (W − F3)× (W − F3)

❏♦❣❛❞♦r B✿ ❡s❝♦❧❤❡ x4 ∈ DF2[x3] ❝♦♠ x4 6= x3

✳✳✳

♥✮ ❏♦❣❛❞♦r A✿ ❡s❝♦❧❤❡ ω((x1, x2, ..., xn) = DFn= Fn×Fn∪(W−Fn)×(W−Fn)

❏♦❣❛❞♦r B✿ ❡s❝♦❧❤❡ xn+1 ∈ DFn−1[xn] ❝♦♠ xn+1 6= xn

✳✳✳

❙✉♣♦♥❤❛ q✉❡ F1 = {x1}, F2 = {x1, x2}, ..., Fn = {x1, x2, ..., xn}✳ ❚❡♠♦s q✉❡✿

✶✮ ❙❡ x1 6= x2✱ ❡♥tã♦ DF1[x2] = X − {x1} ✐♠♣❧✐❝❛ q✉❡ ♥❛ ♣ró①✐♠❛ ❥♦❣❛❞❛ ♦

❥♦❣❛❞♦r B ♥ã♦ ♣♦❞❡ ❡s❝♦❧❤❡r x1✳

✷✮ ❙❡ x1 = x2✱ ❡♥tã♦ DF1[x2] = {x1} ✐♠♣❧✐❝❛ q✉❡ ♥❛ ♣ró①✐♠❛s ❥♦❣❛❞❛ ♦ ❥♦❣❛❞♦r

B ❡st❛ ♦❜r✐❣❛❞♦ ❛ ❡s❝♦❧❤❡r x1✳

❋✐♥❛❧♠❡♥t❡ ♣❛r❛ ♦❜t❡r ✉♠❛ ❡str❛té❣✐❛ ✈❡♥❝❡❞♦r❛ ♥❡st❡ ❥♦❣♦ ♣r♦①✐♠❛❧✱ s✉♣♦♥❤❛q✉❡ F1 = {x1}−{∞}, F2 = {x1, x2}−{∞}, ..., Fn = {x1, x2, ..., xn}−{∞}✳ ❉❡✜♥❛❛ ❡str❛té❣✐❛ ω ❝♦♠♦

ω(x1, x2, ..., xn) = DFn.

❊♥tã♦ q✉❛❧q✉❡r s❡q✉ê♥❝✐❛ ♣r♦①✐♠❛❧ (x1, x2, ..., xn, ...) ❝♦♥✈❡r❣❡ ❛ ∞ ♦✉ é❡✈❡♥t✉❛❧♠❡♥t❡ ❝♦♥st❛♥t❡✳ P♦rt❛♥t♦✱ X é D✲♣r♦①✐♠❛❧✳

✷✾

✸✵ ✸✳✸✳ P❘❖P❘■❊❉❆❉❊❙ ❉❊ ❊❙P❆➬❖❙ P❘❖❳■▼❆▲

✸✳✸ Pr♦♣r✐❡❞❛❞❡s ❞❡ ❊s♣❛ç♦s Pr♦①✐♠❛❧

◆❡st❛ s❡çã♦✱ ♣r♦✈❛r❡♠♦s ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s ❜ás✐❝❛s ❞❡ ❡s♣❛ç♦s ♣r♦①✐♠❛❧❀❝♦♠♦ s✉❜✲❡s♣❛ç♦ ♣r♦①✐♠❛❧✱ ❝❛r❛❝t❡r✐③❛çã♦ ❞❡ ❡s♣❛ç♦ ♠étr✐❝♦ ❝♦♠♣❧❡t♦ ❡♠ ❢✉♥çã♦❞♦ ❡s♣❛ç♦ ❛❜s♦❧✉t❛♠❡♥t❡ ♣r♦①✐♠❛❧ ❡ ♣r♦❞✉t♦ ❡♥✉♠❡rá✈❡❧ ❞❡ ❡s♣❛ç♦s ♣r♦①✐♠❛❧❝♦♠ ❛ ✉♥✐❢♦r♠✐❞❛❞❡ ♣r♦❞✉t♦✳

Pr♦♣♦s✐çã♦ ✸✳✶✹✳ ❙❡ X é ✉♠ ❡s♣❛ç♦ D✲♣r♦①✐♠❛❧ ❡ F é ✉♠ s✉❜❝♦♥❥✉♥t♦ ❢❡❝❤❛❞♦❞❡ X✱ ❡♥tã♦ F é ✉♠ ❡s♣❛ç♦ DF ✲♣r♦①✐♠❛❧✳

❉❡♠♦♥str❛çã♦✳ ❙✉♣♦♥❤❛ X ✉♠ ❡s♣❛ç♦ D✲♣r♦①✐♠❛❧✱ ❡♥tã♦ ❡①✐st❡ ✉♠❛ ❡str❛té❣✐❛D✲✈❡♥❝❡❞♦r❛ ω : A −→ D✳ P❡❧♦ ❊①❡♠♣❧♦ ✷✳✹✱ ❛ ❢❛♠í❧✐❛✿

DF = {D ∩ (F × F ) : D ∈ D},

é ✉♠❛ ✉♥✐❢♦r♠✐❞❛❞❡ ❡♠ F ✳ ❉❡✜♥❛✲s❡ ❛ ❡str❛té❣✐❛ DF ✲✈❡♥❝❡❞♦r❛ ωF : A −→ DF

❞❛❞❛ ♣♦r✿

ωF (x1, x2, ..., xn) = ω(x1, x2, ..., xn) ∩ (F × F )

t❛❧ q✉❡✿

✶✮ ωF (∅) = F × F ✳

✷✮ (x1, x2, ..., xn) s❡❥❛ ✉♠❛ s❡q✉ê♥❝✐❛ ❛❞♠✐ssí✈❡❧ ❝♦♠ (x1, x2..., xn) ∈ F ✳ ❉❡s❞❡q✉❡ ω é ❡str❛té❣✐❛ D✲✈❡♥❝❡❞♦r❛✱ xn+1 ∈ ω(x1, x2, ..., xn−1)[xn] ❡ xn+1 ∈(F × F )[xn]✳ ❈♦♠♦

(ω(x1, x2, ..., xn−1) ∩ (F × F ))[xn] = ω(x1, ..., xn−1)[xn] ∩ (F × F )[xn],

❡♥tã♦ xn+1 ∈ ωF (x1, ..., xn−1)[xn]✳

✸✮ (x1, x2, ..., xn) s❡❥❛ ✉♠❛ s❡q✉ê♥❝✐❛ ❛❞♠✐ssí✈❡❧ ❝♦♠ (x1, x2, ..., xn) ∈ F ✳ ❉❡s❞❡q✉❡ ω é ❡str❛té❣✐❛ D✲✈❡♥❝❡❞♦r❛✱ ω(x1, x2, ..., xn) ⊆ ω(x1, x2, ..., xn−1)✱ ❡♥tã♦

ω(x1, x2, ..., xn) ∩ (F × F ) ⊆ ω(x1, x2, ..., xn−1) ∩ (F × F ).

❆ss✐♠ ωF (x1, x2, ..., xn) ⊆ ωF (x1, x2, ..., xn−1)

✹✮ (x1, x2, ..., xn) s❡❥❛ ✉♠❛ s❡q✉ê♥❝✐❛ ❛❞♠✐ssí✈❡❧ t❛❧ q✉❡ (x1, x2, ..., xn) ∈ F ❡s✉♣♦♥❤❛ q✉❡

⋂

n∈N

ω(x1, x2, ..., xn−1)[xn] 6= ∅✱ ❡♥tã♦ ❡①✐st❡ z ∈ X t❛❧ q✉❡ ❛

s❡q✉ê♥❝✐❛ ♣r♦①✐♠❛❧ (x1, x2, ..., xn, ...) ❝♦♥✈❡r❣❡ ❛ z✱ ❛ss✐♠ z ∈ F ♣♦✐s F é❢❡❝❤❛❞♦✳

P♦rt❛♥t♦✱ F é ✉♠ ❡s♣❛ç♦ DF ✲♣r♦①✐♠❛❧✳

❆ ❝♦♥t✐♥✉❛çã♦✱ ✈❡♠♦s q✉❡ ♦ ❥♦❣♦ ♣r♦①✐♠❛❧ ❞❡✜♥❡ ✉♠ ❡s♣❛ç♦ ♣r♦①✐♠❛❧ q✉❡❝❛r❛❝t❡r✐③❛ ❡s♣❛ç♦ ♠étr✐❝♦ ❝♦♠♣❧❡t♦✱ ❜❛s✐❝❛♠❡♥t❡ é ✉♠❛ ♦✉tr❛ ❢♦r♠❛ ❞❡ ❞✐③❡r❡s♣❛ç♦ ♠étr✐❝♦ ❝♦♠♣❧❡t♦✳

✸✵

✸✶ ✸✳✸✳ P❘❖P❘■❊❉❆❉❊❙ ❉❊ ❊❙P❆➬❖❙ P❘❖❳■▼❆▲

❚❡♦r❡♠❛ ✸✳✶✺✳ ❙❡❥❛ (X, d) ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦✳ ❯♠ ❡s♣❛ç♦ Dd✲♣r♦①✐♠❛❧ (X,Dd)é ❛❜s♦❧✉t❛♠❡♥t❡ D✲♣r♦①✐♠❛❧ s❡✱ ❡ s♦♠❡♥t❡ s❡ (X, d) é ❝♦♠♣❧❡t♦✳

❉❡♠♦♥str❛çã♦✳ ❙❡❥❛ (X,Dd) ✉♠ ❡s♣❛ç♦ ❛❜s♦❧✉t❛♠❡♥t❡ Dd✲♣r♦①✐♠❛❧✱ ♦♥❞❡ Dd é✉♠❛ ❜❛s❡ ✉♥✐❢♦r♠✐❞❛❞❡ ❢♦r♠❛❞♦ ♣❡❧♦s ❡❧❡♠❡♥t♦s ❞❛ ❢♦r♠❛✿

Dn = {(x, y) ∈ X ×X : d(x, y) < 2−n}.

▼♦str❡♠♦s q✉❡ (X, d) é ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦ ❝♦♠♣❧❡t♦✳ ❊①✐st❡ ✉♠❛ ❡str❛té❣✐❛❛❜s♦❧✉t❛♠❡♥t❡ Dd✲✈❡♥❝❡❞♦r❛ ω : A −→ Dd ❞❛❞❛ ♣♦r ω(x1, ..., xn) = Dn+1 t❛❧ q✉❡✿

✶✮ ω(∅) = Dk0 ❀

✷✮ xn+1 ∈ ω(x1, ..., xn−1)[xn] = Dn[xn]❀

✸✮ ω(x1, ..., xn) = Dn ⊂ ω(x1, ..., xn−1) = Dn−1❀

✹✮ ❆ s❡q✉ê♥❝✐❛ ♣r♦①✐♠❛❧ (x1, ..., xn, ...) ❝♦♥✈❡r❣❡ ❡♠ X✳

❙✉♣♦♥❤❛ q✉❡ (X, d) é ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦ ♥ã♦ ❝♦♠♣❧❡t♦✱ ✐st♦ é✿ ❊①✐st❡ ✉♠❛s❡q✉ê♥❝✐❛ ❞❡ ❈❛✉❝❤② (x1, ..., xn, ...) ❡♠ X ❝♦♠ s✉❜s❡q✉ê♥❝✐❛ (y1, ..., yn, ...) q✉❡♥ã♦ ❝♦♥✈❡r❣❡✳ ❉❡s❞❡ q✉❡ ω(∅) = Dk0 ✱ ♦❜s❡r✈❡✲s❡ q✉❡ ❛ s✉❜s❡q✉ê♥❝✐❛ ❞❡✜♥✐❞❛ ♣♦r(y1 = xk0 , y2 = xk1 , ..., yn = xkn−1

, ...) é ✉♠❛ s❡q✉ê♥❝✐❛ ♣r♦①✐♠❛❧✳ ❉❡ ❢❛t♦✿

✶✮ ω(∅) = Dk0 ✱ ❝♦♠ k0 ∈ N

✷✮ ❆s ♣r✐♠❡✐r❛s ❥♦❣❛❞❛s sã♦ Dk0 ❡ xk0 ✳ ❊♥tã♦ ❡①✐st❡ k0 > 0 t❛✐s q✉❡d(xk0 , xk1) < 2−k0 ✱ ♣❛r❛ t♦❞♦ k0, k1 ≥ k0✳ ❖ q✉❡ ✐♠♣❧✐❝❛ q✉❡ xk1 ∈ Dk0 [xk0 ]✳❈♦♥t✐♥✉❛❞♦ ❝♦♠ ❛ ✐♥❞✉çã♦ t❡♠✲s❡✿

xkn ∈ ω(xk0 , xk1 , ..., xkn−2)[xkn−1

] = Dkn−1[xkn−1

].

❆ss✐♠ yn+1 ∈ ω(y1, y2, ..., yn−1)[yn] = Dn[yn]✳

✸✮ ❉❡s❞❡ q✉❡ Kn−1 < Kn✱ ❝♦♠ Kn−1, Kn ∈ N✱ ❡♥tã♦ Dkn ⊆ Dkn−1✳ ❖ q✉❡

✐♠♣❧✐❝❛ q✉❡ ω(y1, y2, ..., yn) ⊆ ω(y1, y2, ..., yn−1)✳

✹✮ ❈♦♠♦ (y1, y2, ..., yn, ...) é ✉♠❛ s❡q✉ê♥❝✐❛ ♣r♦①✐♠❛❧ ❡ ω é ✉♠❛ ❡str❛té❣✐❛❛❜s♦❧✉t❛♠❡♥t❡ Dd✲✈❡♥❝❡❞♦r❛✱ ❡♥tã♦ ❛ s❡q✉ê♥❝✐❛ (y1, y2, ..., yn, ...) ❝♦♥✈❡r❣❡❡♠ X✳

❈♦♠♦ s✉♣♦♠♦s q✉❡ ❛ s❡q✉ê♥❝✐❛ (y1, y2, ..., yn, ...) ♥ã♦ ❝♦♥✈❡r❣❡ ❡ ♣r♦✈❛♠♦s q✉❡❝♦♥✈❡r❣❡ ❡♥tã♦ é ✉♠❛ ❝♦♥tr❛❞✐çã♦✳ P♦rt❛♥t♦✱ (X, d) é ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦❝♦♠♣❧❡t♦✳

❆❣♦r❛ s✉♣♦♥❤❛♠♦s q✉❡ (X, d) ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦ ❝♦♠♣❧❡t♦ ❡ (X,Dd) ✉♠❡s♣❛ç♦ ✉♥✐❢♦r♠❡✳ ▼♦str❡♠♦s q✉❡ (X,Dd) é ✉♠ ❡s♣❛ç♦ ❛❜s♦❧✉t❛♠❡♥t❡ Dd✲♣r♦①✐♠❛❧✳ ❖❜s❡r✈❡✲s❡ q✉❡ ω : A → Dd é ✉♠❛ ❡str❛té❣✐❛ ❛❜s♦❧✉t❛♠❡♥t❡ Dd✲✈❡♥❝❡❞♦r❛ ❞❡✜♥✐❞❛ ♣♦r✿

ω(x1, x2, ..., xn) = Dn+1.

✸✶

✸✷ ✸✳✸✳ P❘❖P❘■❊❉❆❉❊❙ ❉❊ ❊❙P❆➬❖❙ P❘❖❳■▼❆▲

❉❡ ❢❛t♦✱ ♣❡❧❛ Pr♦♣♦s✐çã♦ ✸✳✶✶ ♦❜t❡♠♦s q✉❡ ω é ✉♠❛ ❡str❛té❣✐❛✳ ❊ ❞❡s❞❡ q✉❡(x1, x1, ..., xn, ...) é ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❈❛✉❝❤② ❡♠ X✱ ❡ ✉♠❛ s❡q✉ê♥❝✐❛ ♣r♦①✐♠❛❧♥♦ ❥♦❣♦ ❛❜s♦❧✉t❛♠❡♥t❡ ♣r♦①✐♠❛❧✳ ❊♥tã♦ ❛ s❡q✉ê♥❝✐❛ (x1, x1, ..., xn, ...) ❝♦♥✈❡r❣❡❡♠ X✱ ♣♦✐s (X, d) é ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦ ❝♦♠♣❧❡t♦✳

P♦rt❛♥t♦✱ X é ✉♠ ❡s♣❛ç♦ ❛❜s♦❧✉t❛♠❡♥t❡ Dd✲♣r♦①✐♠❛❧✳

❋✐♥❛❧♠❡♥t❡✱ ♣r♦✈❛r❡♠♦s q✉❡ ♦ ♣r♦❞✉t♦ ❡♥✉♠❡rá✈❡❧ ❞❡ ❡s♣❛ç♦s ♣r♦①✐♠❛❧ é✉♠ ❡s♣❛ç♦s ♣r♦①✐♠❛❧✳ ◆❡st❡ ❝❛s♦✱ ♣r❡❝✐s❛r❡♠♦s ❞❛ ✉♥✐❢♦r♠✐❞❛❞❡ ♣r♦❞✉t♦ q✉❡♣♦❞❡♠♦s ✈❡r ♥❛ ❖❜s❡r✈❛çã♦ ✷✳✽✱ ♣❛r❛ ♦ ❝❛s♦ ❞❡ ♣r♦❞✉t♦ ❡♥✉♠❡rá✈❡❧✳

❚❡♦r❡♠❛ ✸✳✶✻✳ ❙❡ X1, X2, ..., Xn, ... é ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❡s♣❛ç♦s ♣r♦①✐♠❛❧✱ ❡♥tã♦∏

n∈N

Xn é ✉♠ ❡s♣❛ç♦ ♣r♦①✐♠❛❧✳

❉❡♠♦♥str❛çã♦✳ ❙✉♣♦♥❤❛ q✉❡ X1, X2, ..., Xn, ... é ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❡s♣❛ç♦s♣r♦①✐♠❛❧✱ ❡ D1,D2, ...,Dn, ... ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ✉♥✐❢♦r♠✐❞❛❞❡s✳ ❊♥tã♦ ♣❛r❛ t♦❞♦n ∈ N ❡①✐st❡♠ ❡str❛té❣✐❛s Dn✲✈❡♥❝❡❞♦r❛s ωn : An −→ Dn✳ ❈♦♥s✐❞❡r❡✲s❡ ♣❛r❛t♦❞♦ n ∈ N ❡ t♦❞♦ D ∈ Dn ❛ ❢❛♠í❧✐❛ E ❢♦r♠❛❞❛ ♣❡❧♦s ❡❧❡♠❡♥t♦s ❞❛ ❢♦r♠❛✿

ω(x1, x2, ..., xn) = {(x, y) ∈∏

n∈N

Xn ×∏

n∈N

Xn : (x(n), y(n)) ∈ D = ωn(x1, x2, ..., xn)}

❉❛ ❢♦r♠❛ ❝♦♠♦ ❡st❛ ❞❡✜♥✐❞❛ ❝❛❞❛ ❡❧❡♠❡♥t♦ ❞❡ E✱ ❡♥tã♦ é ✉♠❛ s✉❜✲❜❛s❡ ♣❛r❛❛ ✉♥✐❢♦r♠✐❞❛❞❡ ♣r♦❞✉t♦ ❡♠

∏

n∈N

Xn✳ P❛r❛ ❛ ❝♦♥str✉çã♦ ❞❛ ❡str❛té❣✐❛ ✈❡♥❝❡❞♦r❛

ω : A −→ D✱ ❝♦♠❡ç❛♠♦s ♦ ❥♦❣♦ ♣r♦①✐♠❛❧ ❡♥tr❡ ♦s ❥♦❣❛❞♦r❡s A ❡ B ❝♦♠ ❛ss❡❣✉✐♥t❡s ❡s❝♦❧❤❛s✿

✶✮ ❏♦❣❛❞♦r A✿ ❡s❝♦❧❤❡ ω(∅) =∏

n∈N

Xn ×∏

n∈N

Xn❀

❏♦❣❛❞♦r B✿ ❡s❝♦❧❤❡ x1 = (x1(1), x1(2), ..., x1(n), ...) ∈∏

n∈N

Xn❀

✷✮ ❏♦❣❛❞♦r A✿ ❡s❝♦❧❤❡ ω(x1) = ωα(x1(α))✱ ♦♥❞❡ α t♦♠❛ ✈❛❧♦r❡ ❞❡ F1 = {f1(1)}❡ C1 = {f1(1), f1(2), ...}❀❏♦❣❛❞♦r B✿ ❡s❝♦❧❤❡ x2 = (x2(1), x2(2), ..., x2(n), ...) ∈

∏

n∈N

Xn[x1]❀

✸✮ ❏♦❣❛❞♦r A✿ ❡s❝♦❧❤❡ ω(x1, x2) =⋂

α∈F2

ωα(x1(α), x2(α))✱ ♦♥❞❡ α t♦♠❛ ✈❛❧♦r❡s

❞❡ F2 = {f1(1), f1(2), f2(1), f2(2)} ❡ C2 = {f2(1), f2(2), ...}❀❏♦❣❛❞♦r B✿ ❡s❝♦❧❤❡ x3 = (x3(1), x3(2), ..., x3(n), ...) ∈ ω(x1)[x2]❀

✳✳✳

♥✮ ❏♦❣❛❞♦r A✿ ❡s❝♦❧❤❡ ω(x1, x2, ..., xn) =⋂

α∈Fn

ωα(x1(α), x2(α), ..., xn(α))✱ ♦♥❞❡

α t♦♠❛ ✈❛❧♦r❡s ❞❡ Fn = {fi(j) : i ≤ n ❡ j ≤ n} ❡ Cn = {fn(1), fn(2), ...}❀❏♦❣❛❞♦r B✿ ❡s❝♦❧❤❡ xn+1 = (xn+1(1), xn+1(2), ..., xn+1(n), ...) ∈ω(x1, x2, ..., xn−1)[xn]❀

✳✳✳

✸✷

✸✸ ✸✳✸✳ P❘❖P❘■❊❉❆❉❊❙ ❉❊ ❊❙P❆➬❖❙ P❘❖❳■▼❆▲

❚❡♥❞♦ ❞❡s❡♥✈♦❧✈✐❞♦ ♦ ❥♦❣♦ ♣r♦①✐♠❛❧✱ ❛❣♦r❛ ♣❛ss❛♠♦s ❛ ❞❡✜♥✐r ❛ ❡str❛té❣✐❛

ω : A −→ D

❞❛❞❛ ♣♦r ω(x1, x2, ..., xn) =⋂

α∈Fn

ωα(x1(α), x2(α), ..., xn(α))✱ ♦♥❞❡ α t♦♠❛ ✈❛❧♦r❡s

❞❡ Fn = {fi(j) : i ≤ n ❡ j ≤ n} ❡ Cn = {fn(1), fn(2), ...}✳

Pr♦✈❡♠♦s q✉❡ ❛ ❡str❛té❣✐❛ ω é ✉♠❛ ❡str❛té❣✐❛ ✈❡♥❝❡❞♦r❛✳ ❖❜s❡r✈❡✲s❡ q✉❡♣❛r❛ t♦❞♦ α ∈

⋃

n∈N

Cn✱ ❞❡s❞❡ q✉❡⋃

n∈N

Fn =⋃

n∈N

Cn✱ ❡①✐st❡ k t❛❧ q✉❡ α ∈

Fk✳ P♦r ♦✉tr❛ ♣❛rt❡✱ ❞❡s❞❡ q✉❡ ωα é ✉♠❛ ❡str❛té❣✐❛ ✈❡♥❝❡❞♦r❛ ♥♦ ❥♦❣♦♣r♦①✐♠❛❧ ❡♠ Xα ❡♥tã♦ ❛ s❡q✉ê♥❝✐❛ (xk(α), xk+1(α), ...) ❝♦♥✈❡r❣❡ ❛❧❣✉♠ y(α) ♦✉⋂

n>k

ωα(xk(α), xk+1(α), ..., xn(α))[xn+1(α)] = ∅✳ ▲♦❣♦✱ s❡ ❡st❛ ✐♥t❡rs❡çã♦ é ✈❛③✐❛

♣❛r❛ ❛❧❣✉♠ α ∈⋃

n∈N

Cn✱ ❡♥tã♦⋂

n∈N

ω(x1, x2, ..., xn)[xn+1] = ∅✳ P♦rt❛♥t♦✱∏

n∈N

Xn é

✉♠ ❡s♣❛ç♦ ♣r♦①✐♠❛❧ ❝♦♠ ❛ ✉♥✐❢♦r♠✐❞❛❞❡ ♣r♦❞✉t♦✳

✸✸

❈❛♣ít✉❧♦ ✹

❏♦❣♦ ❚♦♣♦❧ó❣✐❝♦ ●r✉❡♥❤❛❣❡

◆❡st❡ ❝❛♣ít✉❧♦ ❛♣r❡s❡♥t❛r❡♠♦s ✉♠ ❥♦❣♦ t♦♣♦❧ó❣✐❝♦ ❛♥t✐❣♦ ✐♥tr♦❞✉③✐❞♦ ♣♦r ●✳●r✉❡♥❤❛❣❡ ♣❛r❛ ❞❡✜♥✐r ♦ ❡s♣❛ç♦ ❝❤❛♠❛❞♦ ❞❡ W ✲❡s♣❛ç♦s q✉❡ é ✉♠❛ ❣❡♥❡r❛❧✐③❛çã♦❞♦s ❡s♣❛ç♦ ♣r✐♠❡✐r♦ ❡♥✉♠❡rá✈❡✐s ❬✺❪✳ ❆❧é♠ ❞✐ss♦✱ ✈❡r❡♠♦s ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s❜ás✐❝❛s q✉❡ s❛t✐s❢❛③ ❡st❡ ❡s♣❛ç♦ ❡ t❛♠❜é♠ ❛ ❧✐❣❛çã♦ ❡♥tr❡ ♦s ❥♦❣♦s Pr♦①✐♠❛❧ ❡ ❥♦❣♦❞❡ ●r✉❡♥❤❛❣❡✳

✹✳✶ ❏♦❣♦ ●r✉❡♥❤❛❣❡

◆❡st❛ s❡çã♦ ❛♣r❡s❡♥t❛r❡♠♦s W ✲❡s♣❛ç♦s q✉❡ ❢♦✐ ✐♥tr♦❞✉③✐❞♦ ♣♦r ●✳ ●r✉❡♥❤❛❣❡❡ ♣r♦✈❛r❡♠♦s q✉❡ t♦❞♦ ❡s♣❛ç♦ ♣r♦①✐♠❛❧ é ✉♠ W ✲❡s♣❛ç♦✳ P❛r❛ ❞❡✜♥✐r W ✲❡s♣❛ç♦s❛ ♣r✐♦r✐ t❡♠♦s q✉❡ ❞❡✜♥✐r ♦ s❡❣✉✐♥t❡ ❥♦❣♦ t♦♣♦❧ó❣✐❝♦ ✐♥✜♥✐t♦✳

❆♥t❡s ❞❡ ❡♥✉♥❝✐❛r ❛ ❞❡✜♥✐çã♦ ❞❡ ✉♠ ❥♦❣♦ ●r✉❡♥❤❛❣❡ ♥✉♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦♦✉ s✐♠♣❧❡s♠❡♥t❡ ❥♦❣♦ ●r✉❡♥❤❛❣❡✳ ❖ ❥♦❣♦ ●r✉❡♥❤❛❣❡ s❡rá r❡❛❧✐③❛❞♦ ❡♥tr❡ ♦s❥♦❣❛❞♦r❡s A ❡ B✳ ●r❛✜❝❛♠❡♥t❡ ♣♦❞❡♠♦s ♦❜s❡r✈❛r ❞❛ ♠❛♥❡✐r❛ ♥❛ q✉❛❧ ❡stã♦❡s❝♦❧❤✐❞❛s ❛s ❥♦❣❛❞❛s ❞♦s ❥♦❣❛❞♦r❡s✳

❋✐❣✉r❛ ✹✳✶✿ ❏♦❣♦ ●r✉❡♥❤❛❣❡

❖❜s❡r✈❡✲s❡ q✉❡ ❛s ❡s❝♦❧❤❛s ❞♦ ❥♦❣❛❞♦r A sã♦ ❛❜❡rt♦s ❞❛ t♦♣♦❧♦❣✐❛ ❡ ❛s ❡s❝♦❧❤❛s❞♦ ❥♦❣❛❞♦r B sã♦ ♣♦♥t♦s ❞♦ ❝♦♥❥✉♥t♦ ♦♥❞❡ ❡st❛ ❞❡✜♥✐❞❛ ❛ t♦♣♦❧♦❣✐❛ t❛❧ q✉❡ ❝❛❞❛

✸✹

✸✺ ✹✳✶✳ ❏❖●❖ ●❘❯❊◆❍❆●❊

♣♦♥t♦ ❡s❝♦❧❤✐❞♦ ♣♦r ♦ ❥♦❣❛❞♦r B ❡stá ❝♦♥t✐❞♦ ♥✉♠ ❛❜❡rt♦ ❡s❝♦❧❤✐❞♦ ♣♦r ♦ ❥♦❣❛❞♦rA✱ ❛ss✐♠ ♣❛ss❛♠♦s ❛ ❞❡✜♥✐r ♦ ❥♦❣♦ ●r✉❡♥❤❛❣❡ ❝♦♠♦✿

❉❡✜♥✐çã♦ ✹✳✶✳ ❙❡❥❛♠ X ✉♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦✱ x ✉♠ ❡❧❡♠❡♥t♦ ❞❡ X✳ ❉❡✜♥❛✲s❡❥♦❣♦ ●r✉❡♥❤❛❣❡ ❡♥tr❡ ♦s ❥♦❣❛❞♦r❡s A ❡ B ❡♠ X ♦✉ ❥♦❣♦ ●r✉❡♥❤❛❣❡✱ ❝♦♠♦ ❛s❡s❝♦❧❤❛s ✐♥✜♥✐t❛s ❞♦s ❥♦❣❛❞♦r❡s A ❡ B ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿

✶✮ ❏♦❣❛❞♦r A✿ ❡s❝♦❧❤❡ X❀❏♦❣❛❞♦r B✿ ❡s❝♦❧❤❡ x1 ∈ X❀

✷✮ ❏♦❣❛❞♦r A✿ ❡s❝♦❧❤❡ U1 ✈✐③✐♥❤❛♥ç❛ ❛❜❡rt❛ ❞❡ x❀❏♦❣❛❞♦r B✿ ❡s❝♦❧❤❡ x2 ∈ U1❀

✸✮ ❏♦❣❛❞♦r A✿ ❡s❝♦❧❤❡ U2 ✈✐③✐♥❤❛♥ç❛ ❛❜❡rt❛ ❞❡ x❀❏♦❣❛❞♦r B✿ ❡s❝♦❧❤❡ x3 ∈ U2❀

✳✳✳

♥✮ ❏♦❣❛❞♦r A✿ ❡s❝♦❧❤❡ Un−1 ✈✐③✐♥❤❛♥ç❛ ❛❜❡rt❛ ❞❡ x❀❏♦❣❛❞♦r B✿ ❡s❝♦❧❤❡ xn ∈ Un−1❀

✳✳✳

❆s ❥♦❣❛❞❛s ❞♦s ❥♦❣❛❞♦r❡s A ❡ B ♥❡st❡ ❥♦❣♦ ●r✉❡♥❤❛❣❡ é ✐♥✜♥✐t❛✱ ♦♥❞❡ ♦ ❝♦♥❥✉♥t♦❞❡ ❥♦❣❛❞❛s ❡s❝♦❧❤✐❞❛s ♣❡❧♦s ❥♦❣❛❞♦r❡s sã♦ ❝❤❛♠❛❞❛s ❞❡ ❡str❛té❣✐❛s✳

❯♠❛ ✈❡③ ❞❡✜♥✐❞♦ ❢♦r♠❛❧♠❡♥t❡ ♦ ❥♦❣♦ ●r✉❡♥❤❛❣❡ ❡♥tr❡ ♦s ❥♦❣❛❞♦r❡s A ❡ B ♥✉♠❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ ♦✉ s✐♠♣❧❡s♠❡♥t❡ ❥♦❣♦ ♣r♦①✐♠❛❧✱ ❞❡✜♥✐r❡♠♦s s♦❜ q✉❡ ❝♦♥❞✐çõ❡s♦ ❥♦❣❛❞♦r A ✈❡♥❝❡ ♦ ❥♦❣♦ ●r✉❡♥❤❛❣❡✳ ◆❡st❡ tr❛❜❛❧❤♦✱ ♥ã♦ ❡st❛♠♦s ✐♥t❡r❡ss❛❞♦ss❡ ♦ ❥♦❣❛❞♦r B ✈❡♥❝❡ ♦ ❥♦❣♦ ♣r♦①✐♠❛❧✳ P❛r❛ ♠❛✐♦r❡s r❡❢❡rê♥❝✐❛s s♦❜r❡ ♦ ❥♦❣♦●r✉❡♥❤❛❣❡ s❡ ♦❜s❡r✈❛ ❡♠ ❬✺❪✳

❉❡✜♥✐çã♦ ✹✳✷✳ ❙❡❥❛♠ X ✉♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ ❡ x ✉♠ ❡❧❡♠❡♥t♦ ❞❡ X✳ ❖ ❥♦❣❛❞♦rA ✈❡♥❝❡ ♦ ❥♦❣♦ ●r✉❡♥❤❛❣❡ ❡♠ X ✱ s❡ ❛ s❡q✉ê♥❝✐❛ (x1, x2, ..., xn, ...) ❡s❝♦❧❤✐❞❛ ♣❡❧♦❥♦❣❛❞♦r B ❝♦♥✈❡r❣❡ ❛ x ❡♠ X✳

◆❛ s❡❣✉✐♥t❡ ❞❡✜♥✐çã♦ ❛ ♠❡♥❝✐♦♥❛r✱ ❞❡✜♥✐r❡♠♦s ❛❧❣✉♥s ❡❧❡♠❡♥t♦s ❞♦ ❥♦❣♦●r✉❡♥❤❛❣❡✳

❉❡✜♥✐çã♦ ✹✳✸✳ ❙❡❥❛ X ✉♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦✳ ❈♦♥s✐❞❡r❡ ♦ ❥♦❣♦ ●r✉❡♥❤❛❣❡ ❞❛❉❡✜♥✐çã♦ ✹✳✶

✶✮ ❆ s❡q✉ê♥❝✐❛ ✐♥✜♥✐t❛ (x1, x2, ..., xn, ..) ❢♦r♠❛❞❛ ♣❡❧❛s ❡s❝♦❧❤❛s ❞♦ ❥♦❣❛❞♦r Bé ❝❤❛♠❛❞❛ ❞❡ δ✲s❡q✉ê♥❝✐❛✳

✷✮ ❆ s❡q✉ê♥❝✐❛ ✜♥✐t❛ (U1, x1, U2, x2, ..., Un, xn) ❢♦r♠❛❞❛ ♣❡❧❛s ❡s❝♦❧❤❛s ❞♦s❥♦❣❛❞♦r❡s A ❡ B é ✉♠ ❥♦❣♦ ♣❛r❝✐❛❧ ●r✉❡♥❤❛❣❡ ❞♦ ❥♦❣♦ ●r✉❡♥❤❛❣❡✳

✸✮ ❆ s❡q✉ê♥❝✐❛ ✜♥✐t❛ (x1, x2, ..., xn) ❢♦r♠❛❞❛ ♣❡❧❛s ❡s❝♦❧❤❛s ❞♦ ❥♦❣❛❞♦r B é❝❤❛♠❛❞❛ ❞❡ s❡q✉ê♥❝✐❛ ❛❞♠✐ssí✈❡❧✳

✸✺

✸✻ ✹✳✷✳ W ✲ ❊❙P❆➬❖❙

◆♦t❛çã♦ ✹✳✹✳ ❖ ❝♦♥❥✉♥t♦ ❞❡ t♦❞❛s ❛s s❡q✉ê♥❝✐❛s ❛❞♠✐ssí✈❡✐s ♥♦ ●r✉❡♥❤❛❣❡ é❝❤❛♠❛❞❛ ❝♦♥❥✉♥t♦ ❛❞♠✐ssí✈❡❧ ❡ ❞❡♥♦t❡✲s❡ ♣♦r A✳

❈♦♠♦ ❡♠ t♦❞♦ ❥♦❣♦ ❡①✐st❡♠ ❛s ❡str❛té❣✐❛s❀ ♥❛ ❞❡✜♥✐çã♦ ❞❡ ❥♦❣♦ ●r✉❡♥❤❛❣❡✈✐♠♦s q✉❡ ♦ ❝♦♥❥✉♥t♦ ❞❡ ❡s❝♦❧❤❛s ❞❡ ❝❛❞❛ ❥♦❣❛❞♦r é ✉♠❛ ❡str❛té❣✐❛✳ ❆❣♦r❛❢♦r♠❛❧♠❡♥t❡ ❞❡✜♥✐r❡♠♦s ❡str❛té❣✐❛ ♣❛r❛ ♦ ❥♦❣❛❞♦r A ♥♦ ❥♦❣♦ ●r✉❡♥❤❛❣❡✱ ♣♦r♠❡✐♦ ❞❛s ❡s❝♦❧❤❛s ❞♦ ❥♦❣❛❞♦r A✳

❉❡✜♥✐çã♦ ✹✳✺✳ ❙❡❥❛♠ X ✉♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ ❡ x ✉♠ ❡❧❡♠❡♥t♦ ❞❡ X✳ ❉❡✜♥❛✲s❡❛ ❡str❛té❣✐❛ ♣❛r❛ ♦ ❥♦❣❛❞♦r A ♥♦ ❥♦❣♦ ●r✉❡♥❤❛❣❡ ❝♦♠♦ ✉♠❛ ❢✉♥çã♦✿

δ : A −→ N(x)

t❛❧ q✉❡

✶✮ δ(∅) = X❀

✷✮ xn+1 ∈ δ(x1, x2, ..., xn)✱

♦♥❞❡ A é ♦ ❝♦♥❥✉♥t♦ ❛❞♠✐ssí✈❡❧✱ ❡ N(x) ✉♠ s✐st❡♠❛ ❞❡ ✈✐③✐♥❤❛♥ç❛s ❞❡ x ❡♠ X✳

❇❛s✐❝❛♠❡♥t❡ ♣♦❞❡♠♦s ♦❜s❡r✈❛r q✉❡ ❛s ❝♦♥❞✐çõ❡s ❞♦ ❥♦❣♦ ●r✉❡♥❤❛❣❡ é ❛❡str❛té❣✐❛ ❞♦ ❥♦❣❛❞♦r A✳ P❛r❛ ✜♥❛❧✐③❛r ♣❛ss❛♠♦s ❛ ❞❡✜♥✐r ✉♠❛ ❡str❛té❣✐❛✈❡♥❝❡❞♦r❛ ♣❛r❛ ♦ ❥♦❣❛❞♦r A ♥♦ ❥♦❣♦ ●r✉❡♥❤❛❣❡✳ ❆♥t❡s ❞❡ ♣❛ss❛r ♥❛ s❡❣✉✐♥t❡❞❡✜♥✐çã♦✱ ❛ ♣♦st❡r✐♦r✐ ✈❡r❡♠♦s ❛s ❝♦♥❞✐çõ❡s ❞❡ ✈❡♥❝❡r ♦ ❥♦❣♦ ●r✉❡♥❤❛❣❡✳

❉❡✜♥✐çã♦ ✹✳✻✳ ❙❡❥❛♠ X ✉♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦✱ x ✉♠ ❡❧❡♠❡♥t♦ ❞❡ X ❡ δ : A −→N(x) ✉♠❛ ❡str❛té❣✐❛ ♥♦ ❥♦❣♦ ●r✉❡♥❤❛❣❡✳ δ é ✉♠❛ ❡str❛té❣✐❛ ✈❡♥❝❡❞♦r❛✱ s❡ ❛ δ✲s❡q✉ê♥❝✐❛ (x1, x2, ..., xn, ...) ❝♦♥✈❡r❣❡ ❛ x✱ ♦♥❞❡ A é ♦ ❝♦♥❥✉♥t♦ ❛❞♠✐ssí✈❡❧ ❡ N(x)✉♠ s✐st❡♠❛ ❞❡ ✈✐③✐♥❤❛♥ç❛s ❞❡ x ❡♠ X✳

✹✳✷ W ✲ ❊s♣❛ç♦s

◆❡st❛ s❡çã♦✱ t❡♥❞♦ ❞❡✜♥✐❞♦ ♦ ❥♦❣♦ ●r✉❡♥❤❛❣❡✱ ♣❛ss❛r❡♠♦s ❛ ❞❡✜♥✐r ♦ ❡s♣❛ç♦t♦♣♦❧ó❣✐❝♦ ❝❤❛♠❛❞♦ ❞❡ ❡s♣❛ç♦ ✧W ✲❡s♣❛ç♦ ✧❡ t❛♠❜é♠ ✈❡r❡♠♦s ❛❧❣✉♥s ❡①❡♠♣❧♦s❜ás✐❝♦s✱ ❝♦♠♦❀ ❛ r❡❧❛çã♦ q✉❡ ❡①✐st❡ ❡♥tr❡ ♦s ❥♦❣♦s ♣r♦①✐♠❛❧ ❡ ●r✉❡♥❤❛❣❡✳ ❈♦♠♦❡st❡ ❥♦❣♦ ●r✉❡♥❤❛❣❡ t❛♠❜é♠ ❞❡✜♥✐✉ ❛❧❣✉♥s ♦✉tr♦s ❡s♣❛ç♦s t♦♣♦❧ó❣✐❝♦s ❬✺❪✱ ♠❛s♥♦ss♦ ✐♥t❡r❡ss❡ é só ❞❡✜♥✐r W ✲❡s♣❛ç♦s ♣❛r❛ ❝♦♠♣❛r❛r ❝♦♠ ♦ ❡s♣❛ç♦ ♣r♦①✐♠❛❧✳

❉❡✜♥✐çã♦ ✹✳✼✳ ❯♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ X é ✉♠ W ✲❡s♣❛ç♦✱ s❡ ❡①✐st❡ ✉♠❛❡str❛té❣✐❛ ✈❡♥❝❡❞♦r❛ δ : A −→ N(x) ♣❛r❛ ♦ ❥♦❣❛❞♦r A ❥♦❣♦ ●r✉❡♥❤❛❣❡✳

◆♦ s❡❣✉✐♥t❡ t❡♦r❡♠❛ ✈❡r❡♠♦s ❛ r❡❧❛çã♦ q✉❡ ❡①✐st❡ ❡♥tr❡ ♦ ❥♦❣♦ ♣r♦①✐♠❛❧ ❡ ♦❥♦❣♦ ●r✉❡♥❤❛❣❡❀ q✉❡ ❞✐③ q✉❡ t♦❞♦ ❡s♣❛ç♦ ♣r♦①✐♠❛❧ é ✉♠ W ✲❡s♣❛ç♦✳ ❈❛❜❡ ❛❝❧❛r❛rq✉❡ ❛ t♦♣♦❧♦❣✐❛ ❞♦ ❡s♣❛ç♦ ♦♥❞❡ s❡ r❡❛❧✐③❛ ♦ ❥♦❣♦ ●r✉❡♥❤❛❣❡ é ✉♠❛ t♦♣♦❧♦❣✐❛ q✉❡♣r♦✈❡♠ ❞❡ ✉♠❛ ✉♥✐❢♦r♠✐❞❛❞❡✳

✸✻

✸✼ ✹✳✷✳ W ✲ ❊❙P❆➬❖❙

❚❡♦r❡♠❛ ✹✳✽✳ ❙❡ X é ✉♠ ❡s♣❛ç♦ ♣r♦①✐♠❛❧✱ ❡♥tã♦ X é ✉♠ W ✲❡s♣❛ç♦✳

❉❡♠♦♥str❛çã♦✳ ❙❡❥❛ X ✉♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ ♣r♦①✐♠❛❧✱ (X,D) ✉♠ ❡s♣❛ç♦ D✲♣r♦①✐♠❛❧✱ D ✉♠❛ ❜❛s❡ ✉♥✐❢♦r♠✐❞❛❞❡ ❛❜❡rt❛ s✐♠étr✐❝❛✱ ❡♥tã♦ ❡①✐st❡ ✉♠❛ ❡str❛té❣✐❛D✲✈❡♥❝❡❞♦r❛✱

ω : A −→ D

t❛❧ q✉❡✿

✶✮ ω(∅) = X ×X

✷✮ yn+1 ∈ ω(x, y1, x, y2, x, ..., x, yn, x)[x]

✸✮ ω(x, y1, x, y2, x, ..., x, yn, x) ⊆ ω(x, y1, x, y2, x, ..., x, yn)

✹✮ ❆ s❡q✉ê♥❝✐❛ ♣r♦①✐♠❛❧ (x, y1, x, y2, x, ..., x, yn, x, ...) ❝♦♥✈❡r❣❡ ❡♠ X ♦✉⋂

n∈N

ω((x, y1, x, y2, x, ..., x, yn)[x] = ∅✳

❉❡ ♠❛♥❡✐r❛ ✐♥t✉✐t✐✈❛ ✈❡♠♦s ♦ s❡❣✉✐♥t❡ ❞❡s❡♥❤♦✱ ❛❥✉❞❛ ❛ ✈❡r ❞❡ q✉❡ ♠❛♥❡✐r❛♣♦❞❡♠♦s ❞❡✜♥✐r ✉♠❛ ❡str❛té❣✐❛ ✈❡♥❝❡❞♦r❛ ♥♦ ❥♦❣♦ ♣r♦①✐♠❛❧ ●r✉❡♥❤❛❣❡✳

❋✐❣✉r❛ ✹✳✷✿ Pr♦①✐♠❛❧ ●r✉❡♥❤❛❣❡

❆❣♦r❛ ✜①❡♠♦s x ✉♠ ❡❧❡♠❡♥t♦ ❞❡ X✱ ❡ ❝♦♥s✐❞❡r❡ N(x) ✉♠ s✐st❡♠❛ ❞❡✈✐③✐♥❤❛♥ç❛s ❛❜❡rt❛s ❞❡ x✳ ❉❡✜♥❛✲s❡ ❛ ❡str❛té❣✐❛ ✈❡♥❝❡❞♦r❛ δ : A

′

−→ N(x)♥♦ ❥♦❣♦ ●r✉❡♥❤❛❣❡ ❞❛❞❛ ♣♦r✿

δ(y1, ..., yn) = ω(x, y1, x, ..., x, yn, x)[x]

t❛❧ q✉❡✿

✶✮ ❖❜s❡r✈❡✲s❡ q✉❡ δ(∅) = ω(x)[x] é ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❞❡ x✳

✸✼

✸✽ ✹✳✸✳ P❘❖P❘■❊❉❆❉❊❙ ❉❊ W ✲❊❙P❆➬❖❙

✷✮ ❉❡s❞❡ q✉❡ yn+1 ∈ ω(x, y1, x, ..., x, yn, x)[x]✱ ❡♥tã♦ yn+1 ∈ σ(y1, ..., yn)✳

✸✮ ❙❡❥❛ (x, y1, x, y2, x, ..., x, yn, x, ...) ❛ s❡q✉ê♥❝✐❛ ♣r♦①✐♠❛❧ ♥♦ ❥♦❣♦ ♣r♦①✐♠❛❧✱❡♥tã♦ ❝♦♠♦ ω ♣♦r ❤✐♣ót❡s❡ é ✉♠❛ ❡str❛té❣✐❛ D✲✈❡♥❝❡❞♦r❛ ✐♠♣❧✐❝❛ q✉❡ ❛s❡q✉ê♥❝✐❛ (x, y1, x, y2, x, ..., x, yn, x, ...) é ❝♦♥✈❡r❣❡♥t❡ ❡♥tã♦ ❝♦♥✈❡r❣❡ ❛ x✳P♦r ♦✉tr❛ ♣❛rt❡ ❛ s✉❜s❡q✉ê♥❝✐❛ (y1, y2, ..., yn, ...) é ✉♠❛ δ✲s❡q✉ê♥❝✐❛ ♥♦ ❥♦❣♦●r✉❡♥❤❛❣❡ q✉❡ ❝♦♥✈❡r❣❡ ❛ x✳

P♦rt❛♥t♦✱ X é ✉♠ W ✲❡s♣❛ç♦✳

✹✳✸ Pr♦♣r✐❡❞❛❞❡s ❞❡ W ✲❊s♣❛ç♦s

◆❡st❛ s❡çã♦✱ ♣r♦✈❛r❡♠♦s ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s ❜ás✐❝❛s ❞❡ W ✲❡s♣❛ç♦s✱ ❝♦♠♦W ✲s✉❜✲❡s♣❛ç♦s✱ ✉♠ t❡♦r❡♠❛ ✐♠♣♦rt❛♥t❡ q✉❡ é ❛ ❣❡♥❡r❛❧✐③❛çã♦ ❞❡ ❡s♣❛ç♦s ♣r✐♠❡✐r♦❡♥✉♠❡rá✈❡✐s ❬✺❪ ❡ ✜♥❛❧♠❡♥t❡ ♣r♦❞✉t♦ ❡♥✉♠❡rá✈❡❧ ❞❡ W ✲❡s♣❛ç♦s ❝♦♠ ❛ t♦♣♦❧♦❣✐❛❞❡ ❚②❝❤♦♥♦✛ ✭♦✉ t♦♣♦❧♦❣✐❛ ♣r♦❞✉t♦✮✳

Pr♦♣♦s✐çã♦ ✹✳✾✳ ❙❡ X é ✉♠ W ✲❡s♣❛ç♦ ❡ F é ✉♠ s✉❜✲❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ ❞❡ X✱❡♥tã♦ F é ✉♠ W ✲❡s♣❛ç♦✳

❉❡♠♦♥str❛çã♦✳ ❙❡❥❛♠ F ✉♠ s✉❜❝♦♥❥✉♥t♦ ❞❡ X✱ x ✉♠ ❡❧❡♠❡♥t♦ ❞❡ X t❛❧ q✉❡x ∈ F ❡ X ✉♠ W ✲❡s♣❛ç♦✱ ❡♥tã♦ ❡①✐st❡ ✉♠ ❡str❛té❣✐❛ ✈❡♥❝❡❞♦r❛ δ : A −→ N(x)✳❉❡✜♥❛✲s❡ ❛ ❡str❛té❣✐❛ ✈❡♥❝❡❞♦r❛ δ

′

: A′

−→ N(x) ❞❛❞❛ ♣♦r✿

δ′

(x1, x2, ..., xn) = δ(x1, x2, ..., xn) ∩ F

t❛❧ q✉❡✿

✶✮ δ(∅) = F

✷✮ ❙❡❥❛ (x1, x2, ..., xn) ✉♠❛ s❡q✉ê♥❝✐❛ ❛❞♠✐ssí✈❡❧ t❛❧ q✉❡ (x1, x2, ..., xn, ...) é ✉♠❛s❡q✉ê♥❝✐❛ ❡♠ F ✱ ❡♥tã♦ ❝♦♠♦ xn+1 ∈ δ(x1, x2, ..., xn) ♣♦✐s δ é ✉♠❛ ❡str❛té❣✐❛✈❡♥❝❡❞♦r❛✱ ❛ss✐♠ ✐♠♣❧✐❝❛ q✉❡ xn+1 ∈ δ(x1, x2, ..., xn) ∩ F = δ

′

(x1, x2, ..., xn)

✸✮ ❉❛❞❛ (x1, x2, ..., xn, ...) ❛ δ′

✲s❡q✉ê♥❝✐❛ ♥♦ ❥♦❣♦ ●r✉❡♥❤❛❣❡ ❡ F s✉❜✲❡s♣❛ç♦❞❡ X✳ ❈♦♠♦ δ é ✉♠❛ ❡str❛té❣✐❛ ✈❡♥❝❡❞♦r❛✱ ❡♥tã♦ ❛ δ

′

✲s❡q✉ê♥❝✐❛(x1, x2, ..., xn, ...) ❝♦♥✈❡r❣❡ ❛ x✳

P♦rt❛♥t♦✱ ♦ s✉❜✲❡s♣❛ç♦ ❋ é ✉♠ W ✲❡s♣❛ç♦✳

❖s W ✲❡s♣❛ç♦s ❜❛s✐❝❛♠❡♥t❡ é ✉♠❛ ❣❡♥❡r❛❧✐③❛çã♦ ❞♦s ❡s♣❛ç♦s ♣r✐♠❡✐r♦❡♥✉♠❡rá✈❡✐s✳ ❖ s❡❣✉✐♥t❡ t❡♦r❡♠❛ ♥♦s ❞✐③ q✉❡ t♦❞♦ ❡s♣❛ç♦ ♣r✐♠❡✐r♦ ❡♥✉♠❡rá✈❡❧ é✉♠ W ✲❡s♣❛ç♦✳

❚❡♦r❡♠❛ ✹✳✶✵✳ ❙❡ X é ✉♠ ❡s♣❛ç♦ ♣r✐♠❡✐r♦ ❡♥✉♠❡rá✈❡❧✱ ❡♥tã♦ X é ✉♠ W ✲❡s♣❛ç♦✳

✸✽

✸✾ ✹✳✸✳ P❘❖P❘■❊❉❆❉❊❙ ❉❊ W ✲❊❙P❆➬❖❙

❉❡♠♦♥str❛çã♦✳ ❙✉♣♦♥❤❛ q✉❡ X é ✉♠ ❡s♣❛ç♦ ♣r✐♠❡✐r♦ ❡♥✉♠❡rá✈❡❧ ❡ x ✉♠❡❧❡♠❡♥t♦ ❞❡ X✱ ❡♥tã♦ ❡①✐st❡ ✉♠❛ ❜❛s❡ ❞❡ ✈✐③✐♥❤❛♥ç❛ ❧♦❝❛❧ U ❞❡ x✱ t❛❧ q✉❡✿

U1 ⊇ U2 ⊇ ... ⊇ Un ⊇ ...

▲♦❣♦ ❞❡✜♥❛✲s❡ ✉♠❛ ❡str❛té❣✐❛ ✈❡♥❝❡❞♦r❛ δ : A −→ N(x) t❛❧ q✉❡✿

✶✮ δ(∅) = X❀

✷✮ P❡❧❛ ❞❡✜♥✐çã♦ ❞❡ ❥♦❣♦ ●r✉❡♥❤❛❣❡ ❞❡✜♥❛✲s❡ q✉❡ xn+1 ∈ Un = δ(x1, x2, ..., xn)✱

✸✮ ❙❡❥❛ (x1, x2, ..., xn, ...) ❛ δ✲s❡q✉ê♥❝✐❛ ♥♦ ❥♦❣♦ ●r✉❡♥❤❛❣❡✱ ❡♥tã♦ ❝♦♠♦U1, U2, ..., Un, ... é ✉♠❛ s❡q✉ê♥❝✐❛ ❡♥❝❛✐①❛❞❛ ❞❡ ❛❜❡rt♦s✱ ✐♠♣❧✐❝❛ q✉❡ ❛s❡q✉ê♥❝✐❛ (x1, x2, ..., xn, ...) ❝♦♥✈❡r❣❡ ❛ x✳

P♦rt❛♥t♦✱ X é ✉♠ W ✲❡s♣❛ç♦✳

❆❣♦r❛ ♣r♦✈❛r❡♠♦s q✉❡ ♦ ♣r♦❞✉t♦ ❡♥✉♠❡rá✈❡❧ ❞❡ W ✲❡s♣❛ç♦s ❝♦♠ ❛ t♦♣♦❧♦❣✐❛❚②❝❤♦♥♦✛✱✭t♦♣♦❧♦❣✐❛ ♣r♦❞✉t♦✮ é ✉♠ W ✲❡s♣❛ç♦✳

❚❡♦r❡♠❛ ✹✳✶✶✳ ❙❡ X1, X2, ..., Xn, ... é ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ W ✲❡s♣❛ç♦s✱ ❡♥tã♦∏

n∈N

Xn

é ✉♠ W ✲❡s♣❛ç♦✳

❉❡♠♦♥str❛çã♦✳ ❙✉♣♦♥❤❛ q✉❡ X1, X2, ..., Xn, ... é ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ W ✲ ❡s♣❛ç♦s✱❡♥tã♦ ♣❛r❛ t♦❞♦ Xn ❡①✐st❡ ✉♠❛ ❡str❛té❣✐❛ ✈❡♥❝❡❞♦r❛ δn : A −→ N(xn) ♥♦ ❥♦❣♦●r✉❡♥❤❛❣❡✳

❊♥tã♦✱ ❝♦♠♦ ♣❛r❛ t♦❞♦ n ∈ N ❡①✐st❡♠ ❡str❛té❣✐❛s ✈❡♥❝❡❞♦r❛s δn ❡ t♦❞♦U ∈ N(xn) ❛ ❢❛♠í❧✐❛ N ❢♦r♠❛❞❛ ♣❡❧♦s ❡❧❡♠❡♥t♦s ❞❛ ❢♦r♠❛✿

δn(x1, x2, ..., xn) = {y ∈∏

n∈N

Xn : y(n) ∈ δn(x1, x2, ..., xn)}.

➱ ✉♠❛ s✉❜✲❜❛s❡ ♣❛r❛ ❛ t♦♣♦❧♦❣✐❛ ♣r♦❞✉t♦✳ ❆❣♦r❛ ♣❛r❛ ❛ ❝♦♥str✉çã♦ ❞❛ ❡str❛té❣✐❛✈❡♥❝❡❞♦r❛ δ : A −→ N(x)✱ ❝♦♠❡ç❛♠♦s ♦ ❥♦❣♦ ●r✉❡♥❤❛❣❡ ❡♥tr❡ ♦s ❥♦❣❛❞♦r❡s A ❡B ❝♦♠ ❛s s❡❣✉✐♥t❡s ❡s❝♦❧❤❛s✿

✶✮ ❏♦❣❛❞♦r A✿ ❡s❝♦❧❤❡ δ(∅) =∏

n∈N

Xn

❏♦❣❛❞♦r B✿ ❡s❝♦❧❤❡ x1 = (x1(1), x1(2), ..., x1(n), ...) ∈∏

n∈N

Xn

✷✮ ❏♦❣❛❞♦r A✿ ❡s❝♦❧❤❡ δ(x1) = δα(x1(α))✱ ♦♥❞❡ α t♦♠❛ ✈❛❧♦r❡ ❞❡ F1 = {f1(1)}❡ C1 = {f1(1), f1(2), ...}❏♦❣❛❞♦r B✿ ❡s❝♦❧❤❡ x2 ∈ δ(x1) = δα(x1(α))

✸✮ ❏♦❣❛❞♦r A✿ ❡s❝♦❧❤❡ δ(x1, x2) =⋂

α∈F2

δα(x1(α), x2(α))✱ ♦♥❞❡ α t♦♠❛ ✈❛❧♦r❡s

❞❡ F2 = {f1(1), f1(2), f2(1), f2(2)} ❡ C2 = {f2(1), f2(2), ...}❏♦❣❛❞♦r B✿ ❡s❝♦❧❤❡ x3 = (x3(1), x3(2), ..., x3(n), ...) ∈ ω(x1, x2)

✸✾

✹✵ ✹✳✸✳ P❘❖P❘■❊❉❆❉❊❙ ❉❊ W ✲❊❙P❆➬❖❙

✳✳✳

♥✮ ❏♦❣❛❞♦r A✿ ❡s❝♦❧❤❡ δ(x1, x2, ..., xn) =⋂

α∈Fn

δα(x1(α), x2(α), ..., xn(α))✱ ♦♥❞❡

α t♦♠❛ ✈❛❧♦r❡s ❞❡ Fn = {fi(j) : i ≤ n ❡ j ≤ n} ❡ Cn = {fn(1), fn(2), ...}❏♦❣❛❞♦r B✿ ❡s❝♦❧❤❡ xn+1 = (xn+1(1), xn+1(2), ..., xn+1(n), ...) ∈ω(x1, x2, ..., xn−1)[xn]

✳✳✳

❯♠❛ ✈❡③ ❞❡s❡♥✈♦❧✈✐❞♦ ♦ ❥♦❣♦ ●r✉❡♥❤❛❣❡ ♥♦ ♣♦♥t♦ x = (x(1), x(2), ..., x(n), ..) ∈∏

n∈N

Xn✱ ❡♥tã♦ ♣❛ss❛♠♦s ❛ ❞❡✜♥✐r ❛ ❡str❛té❣✐❛✿

δ : A −→ N(x)

❞❛❞❛ ♣♦r δ(x1, x2, ..., xn) =⋂

α∈Fn

δα(x1(α), x2(α), ..., xn(α)) ♦♥❞❡ α t♦♠❛ ✈❛❧♦r❡s

❞❡ Fn = {fi(j) : i ≤ n ❡ j ≤ n} ❡ Cn = {fn(1), fn(2), ...}

❆❣♦r❛ ♣r♦✈❡♠♦s q✉❡ ❛ ❡str❛té❣✐❛ δ é ✉♠❛ ❡str❛té❣✐❛ ✈❡♥❝❡❞♦r✳ ❖❜s❡r✈❡✲s❡ q✉❡♣❛r❛ t♦❞♦ α ∈

⋃

n∈N

Cn✱ ❞❡s❞❡ q✉❡⋃

n∈N

Fn =⋃

n∈N

Cn✱ ❡①✐st❡ k t❛❧ q✉❡ α ∈ Fk✳ P♦r

♦✉tr❛ ♣❛rt❡ ❞❡s❞❡ q✉❡ δα é ✉♠❛ ❡str❛té❣✐❛ ✈❡♥❝❡❞♦r❛ ♥♦ ❥♦❣♦ ●r✉❡♥❤❛❣❡ ❡♠ Xα✱❡♥tã♦ ♣❡❧❛ Pr♦♣♦s✐çã♦ ✶✳✸✵ ❛ s❡q✉ê♥❝✐❛ (xk(α), xk+1(α), ...) ❝♦♥✈❡r❣❡ ❛ x(α) ♣❛r❛t♦❞♦ α✳ ❆ss✐♠ ❛ s❡q✉ê♥❝✐❛ (x1, x2, ..., xn, ...) ❝♦♥✈❡r❣❡ ❛ x✳

P♦rt❛♥t♦✱∏

n∈N

Xn é ✉♠ W ✲❡s♣❛ç♦ ❝♦♠ ❛ t♦♣♦❧♦❣✐❛ ♣r♦❞✉t♦✳

✹✵

❈❛♣ít✉❧♦ ✺

❈♦♥s❡q✉ê♥❝✐❛s ❚♦♣♦❧ó❣✐❝❛s

◆❡st❛ ❝❛♣ít✉❧♦✱ ❛♣r❡s❡♥t❛r❡♠♦s ♦s r❡s✉❧t❛❞♦s ❞❛ t❡♦r✐❛ ❞❡s❡♥✈♦❧✈✐❞❛ ❛ ♣r✐♦r✐❀♦♥❞❡ ✈❡r❡♠♦s ❝♦♥s❡q✉ê♥❝✐❛s t♦♣♦❧ó❣✐❝❛s ❞♦s ❡s♣❛ç♦s ❞❡✜♥✐❞♦s ♥♦s ❝❛♣ít✉❧♦s❛♥t❡r✐♦r❡s✳ ❆ss✐♠✱ ❝♦♠♦ ❛s ✐♠♣❧✐❝❛çõ❡s ❛ ❝♦❧❡çõ❡s ♠❡t❛✲❝♦♠♣❛❝t❛s✱ ♥♦r♠❛❧ ❡❍❛✉s❞♦r✛✳ ❚❛♠❜é♠ ✈❡r❡♠♦s ❛❧❣✉♥s ❝♦♥tr❛✲❡①❡♠♣❧♦s ❞❡ ❡s♣❛ç♦s q✉❛s❡ ♣r♦①✐♠❛❧✳

✺✳✶ ❏♦❣♦ Pr♦①✐♠❛❧ ❋r❛❝♦

◆❡st❛ s❡çã♦✱ ❛♣r❡s❡♥t❛r❡♠♦s ✉♠ ❝❛s♦ ♣❛rt✐❝✉❧❛r ❞❡ ✉♠ ❥♦❣♦ ♣r♦①✐♠❛❧ ❝❤❛♠❛❞♦❥♦❣♦ k✲♣r♦①✐♠❛❧✱ ♣❛r❛ ❞❡✜♥✐r ❡st❡ ❥♦❣♦ ❡♠ ♠❡♥çã♦ ♥✉♠ ❡s♣❛ç♦ ✉♥✐❢♦r♠❡✱ t♦♠❡♠♦s❡♠ ❝♦♥t❛ q✉❡ ♦ ❥♦❣♦ k✲♣r♦①✐♠❛❧ s❡rá r❡❛❧✐③❛❞♦ ❡♥tr❡ ♦s ❥♦❣❛❞♦r❡s A ❡ B✳

❋✐❣✉r❛ ✺✳✶✿ ❏♦❣♦ ❦✲Pr♦①✐♠❛❧

❖♥❞❡ ❛s ❡s❝♦❧❤❛s ❞♦ ❥♦❣❛❞♦r A sã♦ ❡❧❡♠❡♥t♦s ❞❡ ✉♠❛ ✉♥✐❢♦r♠✐❞❛❞❡ ❡ ❛s❡s❝♦❧❤❛s ❞♦ ❥♦❣❛❞♦r B sã♦ ♣♦♥t♦s ❞♦ ❝♦♥❥✉♥t♦ ♦♥❞❡ ❡st❛ ❞❡✜♥✐❞❛ ❛ ✉♥✐❢♦r♠✐❞❛❞❡✳

✹✶

✹✷ ✺✳✶✳ ❏❖●❖ P❘❖❳■▼❆▲ ❋❘❆❈❖

❉❡✜♥✐çã♦ ✺✳✶✳ ❙❡❥❛ (X,D) ✉♠ ❡s♣❛ç♦ ✉♥✐❢♦r♠❡ ❡ k ✉♠ ♥✉♠❡r♦ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦✳❉❡✜♥❛✲s❡ ❥♦❣♦ k✲♣r♦①✐♠❛❧ ❡♠ (X,D) ❡♥tr❡ ♦s ❥♦❣❛❞♦r❡s A ❡ B ♦✉ ❥♦❣♦ k✲♣r♦①✐♠❛❧ ❝♦♠♦ ❛s ❡s❝♦❧❤❛s ✐♥✜♥✐t❛s ❞♦s ❥♦❣❛❞♦r❡s A ❡ B✿

✶✮ ❏♦❣❛❞♦r A✿ ❡s❝♦❧❤❡ X❀❏♦❣❛❞♦r B✿ ❡s❝♦❧❤❡ x1 ∈ X❀

✷✮ ❏♦❣❛❞♦r A✿ ❡s❝♦❧❤❡ D1 ∈ D❀❏♦❣❛❞♦r B✿ ❡s❝♦❧❤❡ x2 ∈ X❀

✸✮ ❏♦❣❛❞♦r A✿ ❡s❝♦❧❤❡ D2 ∈ D ❝♦♠ D2 ⊆ D1❀❏♦❣❛❞♦r B✿ ❡s❝♦❧❤❡ x3 ∈ kD1[x2]❀

✳✳✳

♥✮ ❏♦❣❛❞♦r A✿ ❡s❝♦❧❤❡ Dn−1 ∈ D ❝♦♠ Dn−1 ⊆ ... ⊆ D2 ⊆ D1❀❏♦❣❛❞♦r B✿ ❡s❝♦❧❤❡ xn ∈ kDn−2[xn−1]❀

✳✳✳

❆s ❥♦❣❛❞❛s ❞♦s ❥♦❣❛❞♦r❡s A ❡ B ♥❡st❡ ❥♦❣♦ k✲♣r♦①✐♠❛❧ é ✐♥✜♥✐t❛✱ ♦♥❞❡ ♦ ❝♦♥❥✉♥t♦❞❡ ❥♦❣❛❞❛s ❡s❝♦❧❤✐❞❛s ♣❡❧♦s ❥♦❣❛❞♦r❡s sã♦ ❝❤❛♠❛❞❛s ❞❡ ❡str❛té❣✐❛s✳

❯♠❛ ✈❡③ ❞❡✜♥✐❞♦ ♦ ❥♦❣♦ k✲♣r♦①✐♠❛❧ ❡♥tr❡ ♦s ❥♦❣❛❞♦r❡s A ❡ B ♥✉♠ ❡s♣❛ç♦✉♥✐❢♦r♠❡ ♦✉ s✐♠♣❧❡s♠❡♥t❡ ❥♦❣♦ k✲♣r♦①✐♠❛❧✱ ❞❡✜♥✐r❡♠♦s s♦❜r❡ q✉❛✐s ❝♦♥❞✐çõ❡s ♦❥♦❣❛❞♦r A ✈❡♥❝❡ ♦ ❥♦❣♦ k✲♣r♦①✐♠❛❧✳

❉❡✜♥✐çã♦ ✺✳✷✳ ❙❡❥❛ X ✉♠ ❡s♣❛ç♦ ✉♥✐❢♦r♠❡✳ ❖ ❥♦❣❛❞♦r A ✈❡♥❝❡ ♦ ❥♦❣♦ kD✲♣r♦①✐♠❛❧✱ s❡ ❛ s❡q✉ê♥❝✐❛ (x1, x2, ..., xn, ...) ❡s❝♦❧❤✐❞❛ ♣❡❧♦ ❥♦❣❛❞♦r B ♥♦ ❥♦❣♦ kD✲♣r♦①✐♠❛❧ ❝♦♥✈❡r❣❡ ❡♠ X✳

◆❡st❡ ❥♦❣♦ k✲♣r♦①✐♠❛❧ ❡①✐st❡♠ ❡❧❡♠❡♥t♦s ❝♦♠♦ ♥❛ ❞❡✜♥✐çã♦ ❞♦ ❥♦❣♦ ♣r♦①✐♠❛❧✸✳✶ ✱ ♣♦r❡♠ ♣❛r❛ s✐♠♣❧✐✜❝❛r ♥ã♦ ♠❡♥❝✐♦♥❛r❡♠♦s ♥♦✈❛♠❡♥t❡✳ ❊♥tã♦ ♣❛ss❛r❡♠♦s ❛❞❡✜♥✐r ❛ ❡str❛té❣✐❛ ♥♦ ❥♦❣♦ k✲♣r♦①✐♠❛❧✳

❉❡✜♥✐çã♦ ✺✳✸✳ ❙❡❥❛♠ X ✉♠ ❡s♣❛ç♦ ✉♥✐❢♦r♠❡ ❡ ω : A −→ D ✉♠❛ ❡str❛té❣✐❛

♣❛r❛ ♦ ❥♦❣❛❞♦r A ♥♦ ❥♦❣♦ k✲♣r♦①✐♠❛❧✳ ❖ ❥♦❣❛❞♦r A q✉❛s❡✲✈❡♥❝❡ ♦ ❥♦❣♦ k✲♣r♦①✐♠❛❧s❡ ❛ s❡q✉ê♥❝✐❛ ♣r♦①✐♠❛❧ (x1, x2, ..., xn, ...) s❛t✐s❢❛③ ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿

✶✮ ❆ s❡q✉ê♥❝✐❛ k✲♣r♦①✐♠❛❧ (x1, x2, ..., xn, ...) ❝♦♥✈❡r❣❡ ❡♠ X✱ ♦✉

✷✮⋂

n∈N

kω(x1, x2, ..., xn−1)[xn] = ∅✱

♦♥❞❡ A é ♦ ❝♦♥❥✉♥t♦ ❛❞♠✐ssí✈❡❧ ❡ D é ❛ ✉♥✐❢♦r♠✐❞❛❞❡ ❡♠ X✳

❉❡✜♥✐❞❛ ✉♠❛ ❡str❛té❣✐❛ k✲✈❡♥❝❡❞♦r❛✱ ♣♦❞❡♠♦s ❞❡✜♥✐r ✉♠ ♥♦✈♦ ❡s♣❛ç♦ q✉❡❝❤❛♠❛r❡♠♦s ❞❡ ❡s♣❛ç♦ k✲♣r♦①✐♠❛❧✳

✹✷

✹✸ ✺✳✶✳ ❏❖●❖ P❘❖❳■▼❆▲ ❋❘❆❈❖

❉❡✜♥✐çã♦ ✺✳✹✳ ❙❡❥❛♠ X ✉♠ ❡s♣❛ç♦ ✉♥✐❢♦r♠❡ ❡ ω : A −→ D ✉♠❛ ❡str❛té❣✐❛ ♣❛r❛♦ ❥♦❣❛❞♦r A ♥♦ ❥♦❣♦ k✲♣r♦①✐♠❛❧✳

✶✮ X é ✉♠ ❡s♣❛ç♦ kD✲♣r♦①✐♠❛❧✱ s❡ ❡①✐st❡ ✉♠❛ ❡str❛té❣✐❛ kD✲✈❡♥❝❡❞♦r❛ ω ♣❛r❛♦ ❥♦❣❛❞♦r A ♥♦ ❥♦❣♦ k✲♣r♦①✐♠❛❧✳

✷✮ ❯♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ X é ✉♠ ❡s♣❛ç♦ k✲♣r♦①✐♠❛❧✱ s❡ ❡①✐st❡ ✉♠❛✉♥✐❢♦r♠✐❞❛❞❡ ❡♠ X q✉❡ ✐♥❞✉③ ❛ t♦♣♦❧♦❣✐❛ ❡♠ X t❛❧ q✉❡ X é ✉♠ ❡s♣❛ç♦kD✲♣r♦①✐♠❛❧✳

❈♦♠♦ ♣r✐♠❡✐r❛ ❝♦♥s❡q✉ê♥❝✐❛ ❞♦ ❥♦❣♦ ♣r♦①✐♠❛❧ ❡ k✲♣r♦①✐♠❛❧ é q✉❡ t♦❞♦ ❡s♣❛ç♦k✲♣r♦①✐♠❛❧ é ♣r♦①✐♠❛❧✱ ❞❡s❞❡ q✉❡ kD ⊇ D ♣❛r❛ t♦❞♦ k ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ ❡ t♦❞❛✈✐③✐♥❤❛♥ç❛ ❞✐❛❣♦♥❛❧ D✳

❖❜s❡r✈❛çã♦ ✺✳✺✳ ❙❡ X é ✉♠ ❡s♣❛ç♦ k✲♣r♦①✐♠❛❧✱ ❡♥tã♦ X é ✉♠❛ ❡s♣❛ç♦ ♣r♦①✐♠❛❧✳

❉❡ ❢❛t♦✿ ❙✉♣♦♥❤❛ X ✉♠ ❡s♣❛ç♦ k✲♣r♦①✐♠❛❧✱ ❡♥tã♦ ❡①✐st❡ ✉♠❛ ❡str❛té❣✐❛kD✲✈❡♥❝❡❞♦r❛ kω : A −→ D✳ ❆❣♦r❛ ❞❡✜♥❛✲s❡ ✉♠❛ ❡str❛té❣✐❛ D✲✈❡♥❝❡❞♦r❛ω : A −→ D ❝♦♠✱

ω(x1, x2, ..., xn) ⊆ kω(x1, x2, ..., xn)

t❛❧ q✉❡✿

✶✮ ω(∅) = kω(∅) = X ×X

✷✮ ❙❡ (x1, x2, ..., xn) ✉♠❛ s❡q✉ê♥❝✐❛ ❛❞♠✐ssí✈❡❧✱ xn+1 ∈ kω(x1, x2, ..., xn−1)[xn]✳P❡❧❛ ❞❡✜♥✐çã♦ ❞❡ ❥♦❣♦ ♣r♦①✐♠❛❧ ❡①✐st❡ ✉♠ ❡❧❡♠❡♥t♦ ω(x1, x2, ..., xn−1)[xn]t❛❧ q✉❡ xn+1 ∈ ω(x1, x2, ..., xn)[xn] ⊆ kω(x1, x2, ..., xn−1)[xn]✳

✸✮ P❛r❛ t♦❞♦ kω(x1, x2, ..., xn) ⊆ kω(x1, x2..., xn−1)✱ ❡①✐st❡♠ ω(x1, x2, ..., xn) ⊆kω(x1, x2, ..., xn) ❡ ω(x1, x2, ..., xn−1) ⊆ kω(x1, x2, ..., xn−1)✳ P❡❧❛ ❞❡✜♥✐çã♦❞❡ ❥♦❣♦ ♣r♦①✐♠❛❧ t❡♠✲s❡ q✉❡✱ ω(x1, x2, ..., xn) ⊆ ω(x1, x2, ..., xn−1)

✹✮ ❙❡❥❛ (x1, x2, ..., xn, ...) ✉♠❛ s❡q✉ê♥❝✐❛ ♣r♦①✐♠❛❧ ❞❡s❞❡ q✉❡ ♣❛r❛ t♦❞♦n ∈ N t❡♠✲s❡ q✉❡ ω(x1, x2, ..., xn−1)[xn] ⊆ kω(x1, x2, ..., xn−1)[xn] ❡⋂

n∈N

kω(x1, x2, ..., xn−1)[xn] = ∅✱ ♣♦✐s kω é ✉♠❛ ❡str❛té❣✐❛ D✲✈❡♥❝❡❞♦r❛ ♥♦

❥♦❣♦ ♣r♦①✐♠❛❧✱ ❡♥tã♦⋂

n∈N

ω(x1, x2, ..., xn−1)[xn] = ∅

P♦rt❛♥t♦✱ ω é ✉♠❛ ❡str❛té❣✐❛ D✲✈❡♥❝❡❞♦r❛✱ ❛ss✐♠ X t❛♠❜é♠ é ✉♠ ❡s♣❛ç♦ D✲♣r♦①✐♠❛❧✳

◆❛ s❡❣✉✐♥t❡ ♣r♦♣♦s✐çã♦ ♣r♦✈❛r❡♠♦s q✉❡ ❛ r❡❝✐♣r♦❝❛ é ✈❡r❞❛❞❡✐r❛✳ ◆♦ s❡♥t✐❞♦q✉❡ ❡①✐st❡ ✉♠ k ♥❛t✉r❛❧ t❛❧ q✉❡ ❛❝♦♥t❡❝❡ ♦ s❡❣✉✐♥t❡ ❧❡♠❛✳

▲❡♠❛ ✺✳✻✳ ❙❡ X é ✉♠ ❡s♣❛ç♦ D✲♣r♦①✐♠❛❧ ❡ k ✉♠ ♥✉♠❡r♦ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦✱ ❡♥tã♦❡①✐st❡ ✉♠❛ ❡str❛té❣✐❛ kD✲✈❡♥❝❡❞♦r❛ kω : A −→ D ♣❛r❛ ♦ ❥♦❣❛❞♦r A ♥♦ ❥♦❣♦k✲♣r♦①✐♠❛❧✳

✹✸

✹✹ ✺✳✶✳ ❏❖●❖ P❘❖❳■▼❆▲ ❋❘❆❈❖

❉❡♠♦♥str❛çã♦✳ ❙✉♣♦♥❤❛ X ✉♠ ❡s♣❛ç♦ D✲♣r♦①✐♠❛❧✱ ❡♥tã♦ ❡①✐st❡ ✉♠❛ ❡str❛té❣✐❛D✲✈❡♥❝❡❞♦r❛ ω : A −→ D✳ ❉❡✜♥❛✲s❡ ✉♠❛ ❡str❛té❣✐❛ D✲✈❡♥❝❡❞♦r❛ kω : A −→ D

❝♦♠✱

kω(x1, x2, ..., xn) ⊆ ω(x1, x2, ..., xn),

t❛❧ q✉❡✿

✶✮ kω(∅) = ω(∅) = X ×X

✷✮ ❙❡ (x1, x2, ..., xn) é ✉♠❛ s❡q✉ê♥❝✐❛ ❛❞♠✐ssí✈❡❧✱ xn+1 ∈ ω(x1, x2, ..., xn−1)[xn]✳P❡❧❛ ❞❡✜♥✐çã♦ ❞❡ ❥♦❣♦ k✲♣r♦①✐♠❛❧ ❡①✐st❡ ✉♠ ❡❧❡♠❡♥t♦ kω(x1, x2, ..., xn−1)[xn]t❛❧ q✉❡ xn+1 ∈ kω(x1, x2, ..., xn)[xn] ⊆ ω(x1, x2, ..., xn−1)[xn]✳

✸✮ P❛r❛ t♦❞♦ ω(x1, x2, ..., xn) ⊆ ω(x1, x2, ..., xn−1)✱ ❡①✐st❡♠ kω(x1, x2, ..., xn) ⊆ω(x1, x2, ..., xn) ❡ kω(x1, x2, ..., xn−1) ⊆ kω(x1, x2, ..., xn−1)✱ ♣❡❧❛ ❞❡✜♥✐çã♦❞❡ ❥♦❣♦ ♣r♦①✐♠❛❧ t❡♠✲s❡ q✉❡✱ kω(x1, x2, ..., xn) ⊆ kω(x1, x2, ..., xn−1)

✹✮ ❙❡❥❛ (x1, x2, ..., xn, ...) ✉♠ s❡q✉ê♥❝✐❛ ♣r♦①✐♠❛❧ ❞❡s❞❡ q✉❡ ♣❛r❛ t♦❞♦ n ∈N t❡♠✲s❡ q✉❡ kω(x1, x2, ..., xn−1)[xn] ⊆ ω(x1, x2..., xn−1)[xn] ❡ ❝♦♠♦⋂

n∈N

ω(x1, x2, ..., xn−1)[xn] = ∅✱ ♣♦✐s ω é ✉♠❛ ❡str❛té❣✐❛ D✲✈❡♥❝❡❞♦r❛ ♥♦ ❥♦❣♦

♣r♦①✐♠❛❧✱ ❡♥tã♦⋂

n∈N

kω(x1, x2, ..., xn−1)[xn] = ∅

P♦rt❛♥t♦✱ kω é ✉♠❛ ❡str❛té❣✐❛ kD✲✈❡♥❝❡❞♦r❛✱ ❛ss✐♠ X t❛♠❜é♠ é ✉♠ ❡s♣❛ç♦ kD✲♣r♦①✐♠❛❧✳

◆❛ s❡❣✉✐♥t❡ ♦❜s❡r✈❛çã♦✱ ❞❛r❡♠♦s ✉♠ ❡①❡♠♣❧♦ ❞❡ ✉♠ ❡s♣❛ç♦ q✉❛s❡ ♣r♦①✐♠❛❧♦ q✉❛❧ ❞❛rá ✐♥❢♦r♠❛çã♦ s♦❜r❡ ♦ ♣r♦❞✉t♦ ❞❡ ❡s♣❛ç♦s q✉❛s❡✲♣r♦①✐♠❛❧✳

❖❜s❡r✈❛çã♦ ✺✳✼✳ ❙❡❥❛ S ❛ ❧✐♥❤❛ ❞❡ ❙♦r❣❡♥❢r❡② ♠✉♥✐❞♦ ❞❛ ❜❛s❡ ✉♥✐❢♦r♠✐❞❛❞❡ D

♣❡❧♦s ❡❧❡♠❡♥t♦s ❞❛ ❢♦r♠❛✿

D = [a, b)× [a, b)

❊♥tã♦ t❡♥❞♦ S ♠✉♥✐❞♦ ❞❡ ✉♠❛ ✉♥✐❢♦r♠✐❞❛❞❡ ❝♦♠❡ç❛♠♦s ♦ ❥♦❣♦ ♣r♦①✐♠❛❧ ❡♥tr❡♦s ❥♦❣❛❞♦r❡s A ❡ B ❝♦♠ ❛s s❡❣✉✐♥t❡ ❡s❝♦❧❤❛s✿

✶✮ ❏♦❣❛❞♦r A✿ ❡s❝♦❧❤❡ D1 = [a1, b1)× [a1, b1)✳❏♦❣❛❞♦r B✿ ❡s❝♦❧❤❡ x1 ∈ S✳

✷✮ ❏♦❣❛❞♦r A✿ ❡s❝♦❧❤❡ D2 = [a2, b2)× [a2, b2)✳❏♦❣❛❞♦r B✿ ❡s❝♦❧❤❡ x2 ∈ D1[x1] = [a1, b1)

✸✮ ❏♦❣❛❞♦r A✿ ❡s❝♦❧❤❡ D3 = [a3, b3)× [a3, b3)✳❏♦❣❛❞♦r B✿ ❡s❝♦❧❤❡ x3 ∈ D2[x2] = [a2, b2)

✳✳✳

♥✮ ❏♦❣❛❞♦r A✿ ❡s❝♦❧❤❡ Dn = [an, bn)× [an, bn)✳❏♦❣❛❞♦r B✿ ❡s❝♦❧❤❡ xn ∈ Dn−1[xn−1] = [an−1, bn−1)

✹✹

✹✺ ✺✳✷✳ ❙❊P❆❘❆➬➹❖ ❊ ❈❖❇❊❘❚❯❘❆

✳✳✳

❉❛s ❡s❝♦❧❤❛s ❞♦s ❥♦❣❛❞♦r❡s A ❡ B t❡♠✲s❡ q✉❡ s❡ n t❡♥❞❡ ❛ ∞ ❡♥tã♦ ❛ |an − bn|t❡♥❞❡ ❛ ✧0✧✳ ❊♥tã♦✱ t❡♠♦s ❛s s❡❣✉✐♥t❡s ♣♦ss✐❜✐❧✐❞❛❞❡s✿

✶✮ ❙❡ ❛s ❡s❝♦❧❤❛s ❞♦ ❥♦❣❛❞♦r B sã♦ ❡✈❡♥t✉❛❧♠❡♥t❡ ♥ã♦ ❞❡❝r❡s❝❡♥t❡ ❡♥tã♦ ♦❥♦❣❛❞♦r A ✈❡♥❝❡ ♦ ❥♦❣♦ ♣r♦①✐♠❛❧ ♣❡❧❛ ❝♦♥❞✐çã♦ ✷✳

✷✮ ❙❡ ❛s ❡s❝♦❧❤❛s ❞♦ ❥♦❣❛❞♦r B ♥ã♦ é ❡✈❡♥t✉❛❧♠❡♥t❡ ♥ã♦ ❞❡❝r❡s❝❡♥t❡ ❡♥tã♦ ❛s❡s❝♦❧❤❛s ❞♦ ❥♦❣❛❞♦r B ♣♦ss✉✐ ✉♠❛ s✉❜s❡q✉ê♥❝✐❛ ❝♦♥✈❡r❣❡♥t❡✳

P♦rt❛♥t♦✱ ♣♦❞❡♠♦s ❡♥❝♦♥tr❛r ✉♠ ❡str❛té❣✐❛ ✈❡♥❝❡❞♦r❛ ❡♠ S ❝♦♠ ❛ ✉♥✐❢♦r♠✐❞❛❞❡D✳

✺✳✷ ❙❡♣❛r❛çã♦ ❡ ❈♦❜❡rt✉r❛

◆❡st❛ s❡çã♦✱ ❝♦♠♦ ❝♦♥s❡q✉ê♥❝✐❛s ❞♦s ❡s♣❛ç♦s ❞❡✜♥✐❞♦s ♣❡❧♦ ❥♦❣♦ ♣r♦①✐♠❛❧✱❣❛♥❤❛♠♦s ❛❧❣✉♥s ❡①❡♠♣❧♦s ♣❛r❛ ❡s♣❛ç♦s ❥á ❝♦♥❤❡❝✐❞♦s ❝♦♠♦✱ ❡s♣❛ç♦s ❝♦❧❡çã♦♥♦r♠❛❧ ❡ ❡s♣❛ç♦s ♠❡t❛✲❝♦♠♣❛❝t♦s ❡♥✉♠❡rá✈❡✐s✳

❖ ♣r✐♠❡✐r♦ r❡s✉❧t❛❞♦ ♥♦s ❞✐③ q✉❡ t♦❞♦ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ ♣r♦①✐♠❛❧ ✸✳✾ é ✉♠❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ ❝♦❧❡çã♦ ♥♦r♠❛❧✳

❚❡♦r❡♠❛ ✺✳✽✳ ❙❡ X é ✉♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ ♣r♦①✐♠❛❧✱ ❡♥tã♦ X é ✉♠❛ ❝♦❧❡çã♦♥♦r♠❛❧✳

❉❡♠♦♥str❛çã♦✳ ❙❡❥❛ X ✉♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ ♣r♦①✐♠❛❧✳ ❊♥tã♦ X é ✉♠ D✲♣r♦①✐♠❛❧✱ ❡ ❛ss✉♠❛ q✉❡ D é ✉♠❛ ❜❛s❡ ✉♥✐❢♦r♠✐❞❛❞❡ ❛❜❡rt❛ ❡ s✐♠étr✐❝❛✳ ❖❜s❡r✈❡♣❡❧♦ ▲❡♠❛ ✺✳✻ ❡①✐st❡ ω : A → D ❡str❛té❣✐❛ q✉❛s❡ 4D✲✈❡♥❝❡❞♦r❛ ♥♦ ❥♦❣♦ 4✲♣r♦①✐♠❛❧✳

❙✉♣♦♥❤❛ F ✉♠❛ ❢❛♠í❧✐❛ ❢❡❝❤❛❞❛ ❞✐s❝r❡t❛ ❞❡ X✳ ❊♥tã♦ ❛♣❧✐❝❛♥❞♦r❡❝✉rs✐✈❛♠❡♥t❡ ❛ ❡str❛té❣✐❛ 4D✲✈❡♥❝❡❞♦r❛ ❡①✐st❡ ✉♠❛ ❢❛♠í❧✐❛ L0, L1, L2, ..., Ln, ...❞❡ ❢❛♠í❧✐❛s ❛❜❡rt❛s ❧♦❝❛❧♠❡♥t❡ ✜♥✐t♦s t❛✐s q✉❡ ♣❛r❛ t♦❞♦ n ∈ N t❡♠✲s❡✿

✶✮ Ln+1 r❡✜♥❛ Ln❀

✷✮ Ln = Rn ∪Qn✱ ❡ Rn = Ln −Qn✱ ♦♥❞❡

Qn = {S ∈ Ln : ∃Fα ∈ F ❝♦♠ S ∩ Fα 6= ∅}.

❉❡ ❢❛t♦✱ ❛♣❧✐❝❛♥❞♦ ♦ s❡❣✉✐♥t❡ ♣r♦❝❡❞✐♠❡♥t♦ ♣♦r ✐♥❞✉çã♦ ✈❡♠♦s q✉❡ ❛ ❢❛♠í❧✐❛L0, L1, L2, ..., Ln, ... é ✉♠❛ ❢❛♠í❧✐❛ ❞❡ r❡✜♥❛♠❡♥t♦s ❛❜❡rt♦ ❧♦❝❛❧♠❡♥t❡ ✜♥✐t♦✳

▲✵✮ ❙❡❥❛ L0 = {X} ❡ R0 = L0✳

✹✺

✹✻ ✺✳✷✳ ❙❊P❆❘❆➬➹❖ ❊ ❈❖❇❊❘❚❯❘❆

▲✶✮ ❙❡❥❛ (x1) ✉♠❛ s❡q✉ê♥❝✐❛ ❛❞♠✐ssí✈❡❧ t❛❧ q✉❡ x1 ∈ F1✳ ❊♥tã♦✱ ♣❡❧❛ Pr♦♣♦s✐çã♦✷✳✷✹ ♣❛r❛ {ω(x1)[x] : x ∈ X} ❡①✐st❡ ✉♠ r❡✜♥❛♠❡♥t♦ ❛❜❡rt♦ ❧♦❝❛❧♠❡♥t❡ ✜♥✐t♦L1 q✉❡ ❝♦❜r❡ X✳ ❙❡❥❛ R1 = L1 −Q1✱ ♦♥❞❡✿

Q1 = {S ∈ L1 : ∃Fα ∈ F ❝♦♠, S ∩ Fα 6= ∅}.

▲✷✮ ❙❡❥❛♠ (x1, x2) ✉♠❛ s❡q✉ê♥❝✐❛ ❛❞♠✐ssí✈❡❧ t❛❧ q✉❡✱ x1 ∈ F1✱ x2 ∈ F2 ❡A1 ∈ R1✳ ❊♥tã♦✱ ♣❡❧❛ Pr♦♣♦s✐çã♦ ✷✳✷✹ ♣❛r❛ {ω(x1, x2)[x] : x ∈ X} ❡①✐st❡ ✉♠r❡✜♥❛♠❡♥t♦ ❛❜❡rt♦ ❧♦❝❛❧♠❡♥t❡ ✜♥✐t♦ T2(A1) q✉❡ ❝♦❜r❡ X✳ ▲♦❣♦ ❝♦♥s✐❞❡r❡L2(A1) = {S ∩A1 : S ∈ T2(A1)} ❡ L2 =

⋃

A1∈R1

L2(A1) ❡ R2 = L2 −Q2✱ ♦♥❞❡✿

Q2 = {S ∈ L2 : ∃Fα ∈ F ❝♦♠, S ∩ Fα 6= ∅}.

▲✸✮ ❙❡❥❛♠ (x1, x2, x3) ✉♠❛ s❡q✉ê♥❝✐❛ ❛❞♠✐ssí✈❡❧ t❛❧ q✉❡ x1 ∈ F1, x2 ∈ F2, x3 ∈F3 ❡ A2 ∈ R2✳ ❊♥tã♦✱ ♣❡❧❛ Pr♦♣♦s✐çã♦ ✷✳✷✹ ♣❛r❛ {ω(x1, x2, x3)[x] : x ∈ X}❡①✐st❡ ✉♠ r❡✜♥❛♠❡♥t♦ ❛❜❡rt♦ ❧♦❝❛❧♠❡♥t❡ ✜♥✐t♦ T3(A2) q✉❡ ❝♦❜r❡ X✳ ▲♦❣♦❝♦♥s✐❞❡r❡ L3(A2) = {S ∩ A2 : S ∈ T3(A2)} ❡ L3 =

⋃

A2∈R2

L3(A2) ❡

R3 = L3 −Q3✱ ♦♥❞❡✿

Q3 = {S ∈ L3 : ∃Fα ∈ F ❝♦♠ S ∩ Fα 6= ∅}.

✳✳✳

▲♥✮ ❙❡❥❛♠ (x1, x2, ..., xn) ✉♠❛ s❡q✉ê♥❝✐❛ ❛❞♠✐ssí✈❡❧ t❛❧ q✉❡✱ x1 ∈ F1, ..., xn ∈ Fn

❡ An−1 ∈ Rn−1✳ ❊♥tã♦ ♣❡❧❛ Pr♦♣♦s✐çã♦ ✷✳✷✹ ♣❛r❛ {ω(x1, ..., xn)[x] : x ∈ X}❡①✐st❡ ✉♠ r❡✜♥❛♠❡♥t♦ ❛❜❡rt♦ ❧♦❝❛❧♠❡♥t❡ ✜♥✐t♦ Tn(An−1) q✉❡ ❝♦❜r❡ X✳ ▲♦❣♦❝♦♥s✐❞❡r❡ Ln(An−1) = {S∩An−1 : S ∈ Tn(An−1)} ❡ Ln =

⋃

An−1∈Rn−1

Ln(An−1)

❡ Rn = Ln −Qn✱ ♦♥❞❡✿

Qn = {S ∈ Ln : ∃Fα ∈ F

❝♦♠ S ∩ Fα 6= ∅}.

❆ss✐♠✱ ♥♦t❡ q✉❡ Ln é ✉♠❛ ❢❛♠í❧✐❛ ❛❜❡rt❛ ❧♦❝❛❧♠❡♥t❡ ✜♥✐t❛✱ ❞❡s❞❡ q✉❡Ln(An−1) é ✉♠ r❡✜♥❛♠❡♥t♦ ❛❜❡rt♦ ❧♦❝❛❧♠❡♥t❡ ✜♥✐t♦ ❞❡ An−1✱ ♣❛r❛ t♦❞♦n ∈ N✱ ♦♥❞❡ An−1 é ✉♠ ❡❧❡♠❡♥t♦ ❞❛ ❢❛♠í❧✐❛ ❛❜❡rt❛ ❧♦❝❛❧♠❡♥t❡ ✜♥✐t♦ Ln−1✳

❆❣♦r❛ ♣r❡❝✐s❛♠♦s ❡♥❝♦♥tr❛r ✉♠❛ ❢❛♠í❧✐❛ ❛❜❡rt❛ ❞✐s❥✉♥t❛ ❞♦✐s ❛ ❞♦✐s✳ ❊♥tã♦✱ s❡❥❛Q =

⋃

n∈N

Qn✱ ♣❛r❛ t♦❞♦ x ∈⋃

F∈F

F ❡①✐st❡ A ∈ Q ❝♦♠ x ∈ A✳

❉❡ ❢❛t♦✱ s✉♣♦♥❤❛ q✉❡ ♥ã♦ ❛❝♦♥t❡❝❡ ✐ss♦✱ ❡♥tã♦ ❡①✐st❡ ✉♠ x ∈⋃

F∈F

F t❛❧ q✉❡

x ∈ A ❡ A ♥ã♦ ♣❡rt❡♥❝❡ Q✳

✶✮ ❉❡s❞❡ q✉❡ L1 é ❝♦❜❡rt✉r❛ ❞❡ X✱ ❡①✐st❡ A1 ∈ L1 ❝♦♠ x ∈ A1✳ ❊♥tã♦✱A1 ∈ R1✱ A1 6∈ Q1 ❡ A1 ⊆ ω(x1)[y1]✱ ♦♥❞❡ x1 ∈ F1 ❡ y1 ∈ X✳

✹✻

✹✼ ✺✳✷✳ ❙❊P❆❘❆➬➹❖ ❊ ❈❖❇❊❘❚❯❘❆

✷✮ ❉❡s❞❡ q✉❡ L2(A1) é ❝♦❜❡rt✉r❛ ❞❡ A1✱ ❡①✐st❡ A2 ∈ L2(A1) ❝♦♠ x ∈ A2✳ ❊♥tã♦A2 ∈ R2✱ A2 6∈ Q2 ❡ t❛♠❜é♠ ❡①✐st❡♠ y2 ∈ X ❡ x2 ∈

⋃

F∈F

F ✱ ♦♥❞❡ x ∈ F2 ❝♦♠

A2 ⊆ ω(x1, x2)[y2] ⊆ 4ω(x1)[x2]✳ ❈♦♥t✐♥✉❛♥❞♦ ♣♦r ✐♥❞✉çã♦ t❡♠✲s❡✿

✳✳✳

♥✮ ❉❡s❞❡ q✉❡ Ln(An−1) é ❝♦❜❡rt✉r❛ ❞❡ An−1✱ ❡①✐st❡ An ∈ Ln(An−1) ❝♦♠x ∈ An✳ ❊♥tã♦ An ∈ Rn✱ An 6∈ Qn ❡ t❛♠❜é♠ ❡①✐st❡ xn ∈

⋃

F∈F

F ❝♦♠

An ⊆ 4ω(x1, ..., xn−1)[xn]✳

▲♦❣♦✱ ♣❛r❛ t♦❞♦ n ∈ N✱ x ∈ An ⊆ 4ω(x1, x2, ..., xn−1)[xn]✱ ❡♥tã♦⋂

i∈N

4ω(x1, ..., xn−1)[xn] 6= ∅✳ ❉❡s❞❡ q✉❡ ω é ✉♠❛ ❡str❛té❣✐❛ D✲✈❡♥❝❡❞♦r❛ ❡♥tã♦ ❛

s❡q✉ê♥❝✐❛ ♣r♦①✐♠❛❧ (x1, x2, ..., xn, ...) ❝♦♥✈❡r❣❡✳ ◆♦ ❡♥t❛♥t♦✱ ✐ss♦ é ✐♠♣♦ssí✈❡❧✱ ✉♠❛✈❡③ q✉❡ ♥ã♦ ❤á ❞♦✐s t❡r♠♦s ❝♦♥s❡❝✉t✐✈♦s ♥❛ s❡q✉ê♥❝✐❛ ♣r♦①✐♠❛❧ (x1, x2, ..., xn, ...)sã♦ ❡❧❡♠❡♥t♦s ❞❡ ✉♠ ♠❡s♠♦ ❝♦♥❥✉♥t♦ ❞❡ F✳

P❛r❛ ✜♥❛❧✐③❛r ❛ ❞❡♠♦♥str❛çã♦ t❡♠♦s q✉❡ ❡♥❝♦♥tr❛r ✉♠❛ ❢❛♠í❧✐❛ ❞❡ ❛❜❡rt♦s❞✐s❥✉♥t♦ ❞♦✐s ❛ ❞♦✐s✳ ❉❡ ❢❛t♦✿

❙❡❥❛♠ Q =⋃

n∈N

Qn ❡ ♣❛r❛ ❝❛❞❛ A ∈ Qn✱ ❞❡✜♥❡✲s❡✿

A∗ = A−⋃

Q∈ Q2

Q ∪⋃

Q∈ Q3

Q ∪ · · · ∪⋃

Q∈ Qn−1

Q.

❊♥tã♦✱ W = {W = A∗ : A ∈ Q} é ✉♠❛ ❢❛♠í❧✐❛ ❧♦❝❛❧♠❡♥t❡ ✜♥✐t❛ ❝♦❜r✐♥❞♦ F✳ P❛r❛❝❛❞❛ α ∈ I ❝♦♥s✐❞❡r❡✿

Hα =⋃

A∈Q∗

A, ♦♥❞❡ A ∈ Q∗ ⇐⇒ ∃Fα ∈ F ❝♦♠ A ∩ Fα 6= ∅;

Gα = Hα −⋃

W∈W

W W ∈ W∗ ⇐⇒ W ∩ Fα 6= ∅.

▲♦❣♦✱ Fα ⊆ Gα ♣❛r❛ t♦❞♦ α ∈ I ❡ ♦s Gα sã♦ ❞✐s❥✉♥t♦s ❞♦✐s ❛ ❞♦✐s✳ ❉❡ ❢❛t♦✱s✉♣♦♥❤❛ q✉❡ α 6= β ❡✿

y ∈ Gα ∩Gβ =⇒ y ∈⋃

Q∈Q

Q;

=⇒ ∃ A ∈ Q ❝♦♠ y ∈ A;

=⇒ ∃ W ∈ W ❝♦♠ y ∈ W ⊆ A;

=⇒ A ∩Gα = ∅ ♦✉ A ∩Gβ = ∅;

=⇒ W ∩Gα = ∅ ♦✉ W ∩Gβ = ∅;

=⇒ y 6∈ Gα ♦✉ y 6∈ Gβ.

P♦rt❛♥t♦✱ X é ✉♠❛ ❝♦❧❡çã♦ ♥♦r♠❛❧✳

✹✼

✹✽ ✺✳✷✳ ❙❊P❆❘❆➬➹❖ ❊ ❈❖❇❊❘❚❯❘❆

❈♦♥t✐♥✉❛♥❞♦ ❝♦♠ ❛s ❝❛r❛❝t❡r✐③❛çõ❡s ♣♦r ♠❡✐♦ ❞❡ ✉♠ ❡s♣❛ç♦ ♣r♦①✐♠❛❧ ❡♠s❡❣✉✐❞❛✱ t❡♠♦s ❛ ❝❛r❛❝t❡r✐③❛çã♦ ♣♦r ♠❡✐♦ ❞❡ ❡s♣❛ç♦ q✉❛s❡ ♣r♦①✐♠❛❧✱ q✉❡ ❞✐③ q✉❡t♦❞♦ ❡s♣❛ç♦ q✉❛s❡ ♣r♦①✐♠❛❧ é ♠❡t❛✲❝♦♠♣❛❝t♦ ❡♥✉♠❡rá✈❡❧ ✶✳✸✹✳ ▼❛✐s ♣r❡❝✐s❛♠❡♥t❡❡♥✉♥❝✐❛♠♦s ❛ ❝♦♥t✐♥✉❛çã♦✳

❚❡♦r❡♠❛ ✺✳✾✳ ❙❡ X é ✉♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ q✉❛s❡ ♣r♦①✐♠❛❧✱ ❡♥tã♦ X é ♠❡t❛✲❝♦♠♣❛❝t♦ ❡♥✉♠❡rá✈❡❧✳

❉❡♠♦♥str❛çã♦✳ ❙❡❥❛ X ✉♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ q✉❛s❡ ♣r♦①✐♠❛❧✳ ❊♥tã♦ X é ✉♠❡s♣❛ç♦ q✉❛s❡ D✲♣r♦①✐♠❛❧✱ ❡ ❛ss✉♠❛ q✉❡ D é ✉♠❛ ❜❛s❡ ✉♥✐❢♦r♠✐❞❛❞❡ ❛❜❡rt❛ ❡s✐♠étr✐❝❛✳ ❖❜❡rs❡r✈❡ ♣❡❧♦ ▲❡♠❛ ✺✳✻ ❡①✐st❡ ω : A → D ❡str❛té❣✐❛ q✉❛s❡ 4D✲✈❡♥❝❡❞♦r❛ ♥♦ ❥♦❣♦ 4✲♣r♦①✐♠❛❧✳

❙✉♣♦♥❤❛ K ✉♠❛ ❢❛♠í❧✐❛ ❞❡ ❝♦♥❥✉♥t♦s ❢❡❝❤❛❞♦s ❞❡ X t❛❧ q✉❡✿

K1 ⊇ K2 ⊇ · · · ⊆ Kn ⊆ · · · ❡⋂

n∈N

Kn = ∅

❊♥tã♦✱ ❛♣❧✐❝❛♥❞♦ r❡❝✉rs✐✈❛♠❡♥t❡ ❛ ❡str❛té❣✐❛ q✉❛s❡ 4D✲✈❡♥❝❡❞♦r❛ ❡①✐st❡ ✉♠❛❢❛♠í❧✐❛ L0, L1, L2, ..., Ln, ... ❞❡ ❢❛♠í❧✐❛s ❛❜❡rt❛s ❧♦❝❛❧♠❡♥t❡ ✜♥✐t♦s t❛✐s q✉❡ ♣❛r❛t♦❞♦ n ∈ N t❡♠✲s❡✿

✶✮ Ln+1 r❡✜♥❛ Ln✳

✷✮ Kn =⋃

n∈N

Ln✱ ❡ Rn = Ln −Qn✱ ♦♥❞❡

Qn = {S ∈ Ln : S ∩Kn+1 = ∅}.

❉❡ ❢❛t♦✱ ❛♣❧✐❝❛♥❞♦ ♦ s❡❣✉✐♥t❡ ♣r♦❝❡❞✐♠❡♥t♦ ♣♦r ✐♥❞✉çã♦ ✈❡♠♦s q✉❡ ❛ ❢❛♠í❧✐❛L0, L1, L2, ..., Ln, ... é ✉♠❛ ❢❛♠í❧✐❛ ❞❡ r❡✜♥❛♠❡♥t♦s ❛❜❡rt♦ ❧♦❝❛❧♠❡♥t❡ ✜♥✐t♦✿

▲✵✮ ❙❡❥❛ L0 = {X} ❡ R0 = L0✳

▲✶✮ ❙❡❥❛ (x1) ✉♠❛ s❡q✉ê♥❝✐❛ ❛❞♠✐ssí✈❡❧ t❛❧ q✉❡ x1 ∈ K1✳ ❊♥tã♦ ♣❡❧❛ Pr♦♣♦s✐çã♦✷✳✷✹ ♣❛r❛ {ω(x1)[x] : x ∈ X} ❡①✐st❡ ✉♠ r❡✜♥❛♠❡♥t♦ ❛❜❡rt♦ ❧♦❝❛❧♠❡♥t❡ ✜♥✐t♦L1 q✉❡ ❝♦❜r❡ X✳ ❙❡❥❛ R1 = L1 −Q1✱ ♦♥❞❡✿

Q1 = {S ∈ L1 : S ∩K2 = ∅}.

▲✷✮ ❙❡❥❛♠ (x1, x2) ✉♠❛ s❡q✉ê♥❝✐❛ ❛❞♠✐ssí✈❡❧ t❛❧ q✉❡ x1 ∈ K1, x2 ∈ K2 ❡A1 ∈ R1✳ ❊♥tã♦✱ ♣❡❧❛ Pr♦♣♦s✐çã♦ ✷✳✷✹ ♣❛r❛ {ω(x1)[x] : x ∈ X} ❡①✐st❡ ✉♠r❡✜♥❛♠❡♥t♦ ❛❜❡rt♦ ❧♦❝❛❧♠❡♥t❡ ✜♥✐t♦ T2(A1) q✉❡ ❝♦❜r❡ X✳ ▲♦❣♦ ❝♦♥s✐❞❡r❡L2(A1) = {S ∩ A1 : S ∈ T2(A1)}✱ L2 =

⋃

A1∈R1

L2(A1) ❡ R2 = L2 −Q2✱ ♦♥❞❡✿

Q2 = {S ∈ L2 : S ∩K3 = ∅}

▲✸✮ ❙❡❥❛♠ (x1, x2, x3) ✉♠❛ s❡q✉ê♥❝✐❛ ❛❞♠✐ssí✈❡❧ t❛❧ q✉❡ x1 ∈ K1, x2 ∈ K2, x3 ∈K3 ❡ A2 ∈ R2✳ ❊♥tã♦✱ ♣❡❧❛ Pr♦♣♦s✐çã♦ ✷✳✷✹ ♣❛r❛ {ω(x1, x2, x3)[x] : x ∈ X}

✹✽

✹✾ ✺✳✷✳ ❙❊P❆❘❆➬➹❖ ❊ ❈❖❇❊❘❚❯❘❆

❡①✐st❡ ✉♠ r❡✜♥❛♠❡♥t♦ ❛❜❡rt♦ ❧♦❝❛❧♠❡♥t❡ ✜♥✐t♦ T3(A2) q✉❡ ❝♦❜r❡ X✳ ▲♦❣♦❝♦♥s✐❞❡r❡ L3(A2) = {S ∩ A2 : S ∈ T3(A2)}✱ L3 =

⋃

A2∈R2

L3(A2) ❡ R3 =

L3 −Q3✱ ♦♥❞❡✿

Q3 = {S ∈ L3 : S ∩K4 = ∅}

✳✳✳

▲♥✮ ❙❡❥❛♠ (x1, x2, ..., xn) ✉♠❛ s❡q✉ê♥❝✐❛ ❛❞♠✐ssí✈❡❧ t❛❧ q✉❡ x1 ∈ K1, ..., xn ∈ Kn

❡ An−1 ∈ Rn−1✳ ❊♥tã♦✱ ♣❡❧❛ Pr♦♣♦s✐çã♦ ✷✳✷✹ ♣❛r❛ {ω(x1, x2, ..., xn)[x] :x ∈ X} ❡①✐st❡ ✉♠ r❡✜♥❛♠❡♥t♦ ❛❜❡rt♦ ❧♦❝❛❧♠❡♥t❡ ✜♥✐t♦ Tn(An−1) q✉❡❝♦❜r❡ X✳ ▲♦❣♦✱ ❝♦♥s✐❞❡r❡ Ln(An−1) = {S ∩ An−1 : S ∈ Tn(An−1)}✱Ln =

⋃

An−1∈Rn−1

Ln(An−1) ❡ Rn = Ln −Qn✱ ♦♥❞❡

Qn = {S ∈ Ln : S ∩Kn+1 = ∅}

❆ss✐♠✱ ♥♦t❡ q✉❡ Ln(An−1) é ✉♠ r❡✜♥❛♠❡♥t♦ ❛❜❡rt♦ ❧♦❝❛❧♠❡♥t❡ ✜♥✐t♦ ❞❡An−1 ♣❛r❛ t♦❞♦ n ∈ N ❝✉❥❛ ✉♥✐ã♦ é An−1✱ ♦♥❞❡ An−1 sã♦ ❡❧❡♠❡♥t♦s ❞❛❢❛♠í❧✐❛ ❛❜❡rt❛ ❧♦❝❛❧♠❡♥t❡ ✜♥✐t❛ Ln−1✳ ❊♥tã♦✱ Ln é ❧♦❝❛❧♠❡♥t❡ ✜♥✐t♦✳ ▲♦❣♦✱Ln é ✉♠ r❡✜♥❛♠❡♥t♦ ❛❜❡rt♦ ❧♦❝❛❧♠❡♥t❡ ✜♥✐t♦ ❞❡ Ln−1 t❛❧ q✉❡ Kn ⊆

⋃

L∈Ln

L

♣❛r❛ t♦❞♦ n ∈ N✳

❆❣♦r❛ ❡♠ s❡❣✉✐❞❛ ♠♦str❡♠♦s q✉❡ ♣❛r❛ ❝❛❞❛ x ∈ X✱ ❡①✐st❡ n ∈ N ❝♦♠ A ∈ Qn ❡x ∈ A✳ ❉❡ ❢❛t♦✱ s✉♣♦♥❤❛ q✉❡ ✐ss♦ ♥ã♦ ❛❝♦♥t❡❝❡✿

✶✮ ❉❡s❞❡ q✉❡ L1 ❝♦❜r❡ X✱ ❡①✐st❡ A1 ∈ L1 ❝♦♠ x ∈ A✱ ❡♥tã♦ A1 6∈ Q1 ❡ A1 ∈ R1✱❛ss✐♠ ❡①✐st❡♠ x1 ∈ K1 ❡ y1 ∈ X t❛❧ q✉❡ A1 ⊆ ω(x1)[y1]✳

✷✮ ❉❡s❞❡ q✉❡ L1(A2) ❝♦❜r❡ A1✱ ❡①✐st❡ A2 ∈ L2 ❝♦♠ x ∈ A✱ ❡♥tã♦ A2 6∈ Q2 ❡A2 ∈ R2✱ ❛ss✐♠ ❡①✐st❡♠ x2 ∈ K2 ❡ y2 ∈ X t❛❧ q✉❡ An ⊆ ω(x1, x2)[y2] ⊆4ω(x1)[x2]✳

✳✳✳

♥✮ ❉❡s❞❡ q✉❡ Ln(An−1) ❝♦❜r❡ An−1✱ ❡①✐st❡ An ∈ Ln ❝♦♠ x ∈ A✱ ❡♥tã♦An 6∈ Qn ❡ An ∈ Rn✱ ❛ss✐♠ ❡①✐st❡♠ xn ∈ Kn ❡ yn ∈ X t❛❧ q✉❡An ⊆ ω(x1, x2, ..., xn)[yn] ⊆ 4ω(x1, x2, ..., xn−1)[xn]✳

▲♦❣♦✱ ♣❛r❛ t♦❞♦ n ∈ N✱ x ∈ An ⊆ 4ω(x1, x2, ..., xn−1)[xn]✱⋂

n∈N

4ω(x1, x2, ..., xn−1)[xn] 6= ∅ ❡ ❝♦♠♦ ω é ✉♠❛ ❡str❛té❣✐❛ q✉❛s❡ D✲✈❡♥❝❡❞♦r❛

❡♥tã♦ ❛ s❡q✉ê♥❝✐❛ ♣r♦①✐♠❛❧ (x1, x2, ..., xn, ...) t❡♠ ✉♠ ♣♦♥t♦ ❞❡ ❛❝✉♠✉❧❛çã♦✳ P♦r♦✉tr❛ ♣❛rt❡✱ xn, xn+1, ... ∈ Kn✱ q✉❛❧q✉❡r ♣♦♥t♦ ❞❡ ❛❝✉♠✉❧❛çã♦ ❞❡ xn, xn+1, ... ❡stát❛♠❜é♠ ❡♠ Kn✳ ❆ss✐♠✱ t♦❞♦s ♦s ♣♦♥t♦s ❞❡ ❛❝✉♠✉❧❛çã♦ ❞❛ s❡q✉ê♥❝✐❛ ♣r♦①✐♠❛❧(x1, x2, ..., xn, ...) ♣❡rt❡♥❝❡♠ ❛

⋂

n∈N

Kn✱ ♦ q✉❡ ❝♦♥tr❛❞✐③ q✉❡⋂

n∈N

Kn = ∅✳

✹✾

✺✵ ✺✳✸✳ ❈❖◆❙❊◗❯✃◆❈■❆ ❚❖P❖▲Ó●■❈❆

P❛r❛ ✜♥❛❧✐③❛r ❛ ❞❡♠♦♥str❛çã♦ t❡♠♦s q✉❡ ❡♥❝♦♥tr❛r ✉♠❛ ❢❛♠í❧✐❛ ❞❡ ❛❜❡rt♦s❡♥❝❛✐①❛❞♦s✳ ❉❡ ❢❛t♦✿

U1 =⋃

L∈L1

L

❡ ♣❛r❛ t♦❞♦ n > 1✱ ❝♦♥s✐❞❡r❡✿

Un =⋃

L∈L1

L−⋃

Q∈ Q2

Q ∪⋃

Q∈ Q3

Q ∪ · · · ∪⋃

Q∈ Qn−1

Q.

❊♥tã♦✱ ♣❛r❛ t♦❞♦ n ∈ N t❡♠✲s❡ q✉❡ Kn ⊆ Un✱ Un+1 ⊆ Un ❡⋂

n∈N

Un = ∅✳ P♦rt❛♥t♦✱

X é ♠❡t❛✲❝♦♠♣❛❝t♦ ❡♥✉♠❡rá✈❡❧✳

✺✳✸ ❈♦♥s❡q✉ê♥❝✐❛ ❚♦♣♦❧ó❣✐❝❛

◆❡st❛ s❡çã♦✱ ❛♣r❡s❡♥t❛r❡♠♦s ♦ s❡❣✉✐♥t❡ t❡♦r❡♠❛ q✉❡ ❞✐③✱ q✉❡ t♦❞♦ ❡s♣❛ç♦q✉❛s❡✲♣r♦①✐♠❛❧ é ✉♠❛ ❝♦❧❡çã♦ ❍❛✉s❞♦r✛ ✶✳✸✻✳

❚❡♦r❡♠❛ ✺✳✶✵✳ ❙❡ X é ✉♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ q✉❛s❡ ♣r♦①✐♠❛❧✱ ❡♥tã♦ X é ✉♠❛❝♦❧❡çã♦ ❍❛✉s❞♦r✛✳

❉❡♠♦♥str❛çã♦✳ ❙❡❥❛ X ✉♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ q✉❛s❡ ♣r♦①✐♠❛❧✳ ❊♥tã♦ X é ✉♠❡s♣❛ç♦ q✉❛s❡ D✲♣r♦①✐♠❛❧✱ ❡ ❛ss✉♠❛ q✉❡ D é ✉♠❛ ❜❛s❡ ✉♥✐❢♦r♠✐❞❛❞❡ ❛❜❡rt❛ ❡s✐♠étr✐❝❛✳ ❖❜s❡r✈❡ ♣❡❧♦ ▲❡♠❛ ✺✳✻ q✉❡ ❡①✐st❡ ω : A → D ❡str❛té❣✐❛ q✉❛s❡ 4D✲✈❡♥❝❡❞♦r❛ ♥♦ ❥♦❣♦ 4✲♣r♦①✐♠❛❧✳

❙✉♣♦♥❤❛ F ✉♠ ❝♦♥❥✉♥t♦ ❢❡❝❤❛❞♦ ❞✐s❝r❡t♦ ❡♠ X✳ ❊♥tã♦✱ ❛♣❧✐❝❛♥❞♦r❡❝✉rs✐✈❛♠❡♥t❡ ❛ ❡str❛té❣✐❛ 4D✲✈❡♥❝❡❞♦r❛ ❡①✐st❡ ✉♠❛ ❢❛♠í❧✐❛ L0, L1, L2, ..., Ln, ...❞❡ ❢❛♠í❧✐❛s ❛❜❡rt❛s ❧♦❝❛❧♠❡♥t❡ ✜♥✐t♦s t❛✐s q✉❡ ♣❛r❛ t♦❞♦ n ∈ N t❡♠✲s❡✿

✶✮ Ln+1 r❡✜♥❛ Ln❀

✷✮ Ln = Rn ∪Qn✱ ❡ Rn = Ln −Qn✱ ♦♥❞❡

Qn = {S ∈ Ln : ∃ xα ∈ F ❝♦♠, S ∩ {xα} 6= ∅}.

❉❡ ❢❛t♦✱ ❛♣❧✐❝❛♥❞♦ ♦ s❡❣✉✐♥t❡ ♣r♦❝❡❞✐♠❡♥t♦ ♣♦r ✐♥❞✉çã♦ ✈❡♠♦s q✉❡ ❛ ❢❛♠í❧✐❛L0, L1, L2, ..., Ln, ... é ✉♠❛ ❢❛♠í❧✐❛ ❞❡ r❡✜♥❛♠❡♥t♦s ❛❜❡rt♦ ❧♦❝❛❧♠❡♥t❡ ✜♥✐t♦✳

▲✵✮ ❙❡❥❛ L0 = {X} ❡ R0 = L0✳

▲✶✮ ❙❡❥❛ (x1) ✉♠❛ s❡q✉ê♥❝✐❛ ❛❞♠✐ssí✈❡❧ t❛❧ q✉❡ x1 ∈ F1✳ ❊♥tã♦✱ ♣❡❧❛ Pr♦♣♦s✐çã♦✷✳✷✹ ♣❛r❛ {ω(x1)[x] : x ∈ X} ❡①✐st❡ ✉♠ r❡✜♥❛♠❡♥t♦ ❛❜❡rt♦ ❧♦❝❛❧♠❡♥t❡ ✜♥✐t♦L1 q✉❡ ❝♦❜r❡ X✳ ❙❡❥❛ R1 = L1 −Q1✱ ♦♥❞❡✿

Q1 = {S ∈ L1 : ∃ xα ∈ F ❝♦♠, S ∩ {xα} 6= ∅}.

✺✵

✺✶ ✺✳✸✳ ❈❖◆❙❊◗❯✃◆❈■❆ ❚❖P❖▲Ó●■❈❆

▲✷✮ ❙❡❥❛♠ (x1, x2) ✉♠❛ s❡q✉ê♥❝✐❛ ❛❞♠✐ssí✈❡❧ t❛❧ q✉❡ x1 ∈ F1✱ x2 ∈ F2 ❡A1 ∈ R1✳ ❊♥tã♦✱ ♣❡❧❛ Pr♦♣♦s✐çã♦ ✷✳✷✹ ♣❛r❛ {ω(x1, x2)[x] : x ∈ X} ❡①✐st❡ ✉♠r❡✜♥❛♠❡♥t♦ ❛❜❡rt♦ ❧♦❝❛❧♠❡♥t❡ ✜♥✐t♦ T2(A1) q✉❡ ❝♦❜r❡ X✳ ▲♦❣♦✱ ❝♦♥s✐❞❡r❡L2(A1) = {S ∩A1 : S ∈ T2(A1)} ❡ L2 =

⋃

A1∈R1

L2(A1) ❡ R2 = L2 −Q2✱ ♦♥❞❡✿

Q2 = {S ∈ L2 : ∃ xα ∈ F ❝♦♠, S ∩ {xα} 6= ∅}.

▲✸✮ ❙❡❥❛♠ (x1, x2, x3) ✉♠❛ s❡q✉ê♥❝✐❛ ❛❞♠✐ssí✈❡❧ t❛❧ q✉❡ x1 ∈ F1, x2 ∈ F2, x3 ∈F3 ❡ A2 ∈ R2✳ ❊♥tã♦ ♣❡❧❛ Pr♦♣♦s✐çã♦ ✷✳✷✹ ♣❛r❛ {ω(x1, x2, x3)[x] : x ∈ X}❡①✐st❡ ✉♠ r❡✜♥❛♠❡♥t♦ ❛❜❡rt♦ ❧♦❝❛❧♠❡♥t❡ ✜♥✐t♦ T3(A2) q✉❡ ❝♦❜r❡ X✳ ▲♦❣♦❝♦♥s✐❞❡r❡ L3(A2) = {S ∩ A2 : S ∈ T3(A2)} ❡ L3 =

⋃

A2∈R2

L3(A2) ❡

R3 = L3 −Q3✱ ♦♥❞❡✿

Q3 = {S ∈ L3 : ∃ xα ∈ F ❝♦♠, S ∩ {xα} 6= ∅}.

✳✳✳

▲♥✮ ❙❡❥❛♠ (x1, x2, ..., xn) ✉♠❛ s❡q✉ê♥❝✐❛ ❛❞♠✐ssí✈❡❧ t❛❧ q✉❡ x1 ∈ F1, ..., xn ∈ Fn

❡ An−1 ∈ Rn−1✳ ❊♥tã♦ ♣❡❧❛ Pr♦♣♦s✐çã♦ ✷✳✷✹ ♣❛r❛ {ω(x1, ..., xn)[x] : x ∈ X}❡①✐st❡ ✉♠ r❡✜♥❛♠❡♥t♦ ❛❜❡rt♦ ❧♦❝❛❧♠❡♥t❡ ✜♥✐t♦ Tn(An−1) q✉❡ ❝♦❜r❡ X✳ ▲♦❣♦❝♦♥s✐❞❡r❡ Ln(An−1) = {S∩An−1 : S ∈ Tn(An−1)} ❡ Ln =

⋃

An−1∈Rn−1

Ln(An−1)

❡ Rn = Ln −Qn✱ ♦♥❞❡✿

Qn = {S ∈ Ln : ∃ xα ∈ F;S ∩ {xα} 6= ∅}.

❆ss✐♠✱ ♥♦t❡ q✉❡ Ln é ✉♠❛ ❢❛♠í❧✐❛ ❛❜❡rt❛ ❧♦❝❛❧♠❡♥t❡ ✜♥✐t❛✱ ❞❡s❞❡ q✉❡Ln(An−1) é ✉♠ r❡✜♥❛♠❡♥t♦ ❛❜❡rt♦ ❧♦❝❛❧♠❡♥t❡ ✜♥✐t♦ ❞❡ An−1✱ ♣❛r❛ t♦❞♦n ∈ N✱ ♦♥❞❡ An−1 é ✉♠ ❡❧❡♠❡♥t♦ ❞❛ ❢❛♠í❧✐❛ ❛❜❡rt❛ ❧♦❝❛❧♠❡♥t❡ ✜♥✐t♦ Ln−1✳

❆❣♦r❛ ♣r❡❝✐s❛♠♦s ❡♥❝♦♥tr❛r ✉♠❛ ❢❛♠í❧✐❛ ❛❜❡rt❛ ❞✐s❥✉♥t❛ ❞♦✐s ❛ ❞♦✐s✳ ❊♥tã♦✱ s❡❥❛Q =

⋃

n∈N

Qn✱ ♣❛r❛ t♦❞♦ x ∈ F ❡①✐st❡ n t❛❧ q✉❡ x ∈⋃

Qn✱ ❞❡ ♦✉tr❛ ❢♦r♠❛ ✐st♦

❞❡✜♥❡ ✉♠❛ s❡q✉ê♥❝✐❛ ♣r♦①✐♠❛❧ (x1, x2, ..., xn, ...) ❝♦♥t✐❞❛ ❡♠ F q✉❡ t❡♠ ✉♠ ♣♦♥t♦❞❡ ❛❝✉♠✉❧❛çã♦✱ ❝♦♥tr❛❞✐③❡♥❞♦ q✉❡ F é ✉♠ ❝♦♥❥✉♥t♦ ❞✐s❝r❡t♦✳ ❆ss✐♠ Q ♣♦ss✉✐ ✉♠r❡✜♥❛♠❡♥t♦ ❛❜❡rt♦ ❧♦❝❛❧♠❡♥t❡ ✜♥✐t♦ W✳ ▲♦❣♦ ♣❛r❛ t♦❞♦ xα ∈ F s❡❥❛✿

Hα =⋃

{A ∈ Q t❛❧ q✉❡ xα ∈ A}

❡

Gα = Hα −⋃

{W t❛❧ q✉❡ W ∈ W ❡ xα 6∈ W}

▲♦❣♦✱ xα ∈ Gα ❡ ❛ ❢❛♠í❧✐❛ ❞❡ Gα✬s sã♦ ❞✐s❥✉♥t♦s ❞♦✐s ❛ ❞♦✐s✳ P♦rt❛♥t♦✱ X é ✉♠❛❝♦❧❡çã♦ ❍❛✉s❞♦r✛✳

✺✶

❈♦♥s✐❞❡r❛çõ❡s ❋✐♥❛✐s

❆ ♣❛rt✐r ❞❡st❡ tr❛❜❛❧❤♦✱ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡ ♦s ❡s♣❛ç♦s ✐♥❞✉③✐❞♦s ♣❡❧♦ ❥♦❣♦♣r♦①✐♠❛❧✱ ❝❛r❛❝t❡r✐③❛ ❡s♣❛ç♦ ♠étr✐❝♦ ❝♦♠♣❧❡t♦✱ ❣♦③❛ ❞❡ ❝❡rt❛s ♣r♦♣r✐❡❞❛❞❡s ❡t❡♠ ✉♠❛ r❡❧❛çã♦ ❝♦♠ ♦ ❡s♣❛ç♦ ✐♥❞✉③✐❞♦ ♣❡❧♦ ❥♦❣♦ ●r✉❡♥❤❛❣❡✳ ❆❧❡♠ ❞✐ss♦ t❡♠✐♠♣❧✐❝❛çõ❡s ❛ ❡s♣❛ç♦s ❝♦❧❡çã♦ ♥♦r♠❛❧✱ ❡s♣❛ç♦s ♠❡t❛✲❝♦♠♣❛❝t♦s ❡ ❡s♣❛ç♦s ❝♦❧❡çã♦❍❛✉s❞♦r✛✳

✺✷

❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s

❬✶❪ ❇❊▲▲ ❏✳❘✳ ❚❤❡ ❯♥✐❢♦r♠ ❇♦① Pr♦❞✉❝t✳ Pr♦❝❡❡❞✐♥❣s ♦❢ t❤❡ ❆♠❡r✐❝❛♥▼❛t❤❡♠❛t✐❝❛❧ ❙♦❝✐❡t②✱ ✷✵✶✹✳

❬✷❪ ❇❊▲▲ ❏✳❘✳ ❆♥ ■♥✜♥✐t❡ ●❛♠❡ ✇✐t❤ ❚♦♣♦❧♦❣✐❝❛❧ ❈♦♥s❡q✉❡♥❝❡s✳❚♦♣♦❧♦❣② ❛♥❞ ✐ts ❆♣♣❧✐❝❛t✐♦♥s✱ ✷✵✶✹✳

❬✸❪ ❉❯●❯◆❏■ ❏✳ ❚♦♣♦❧♦❣②✳ ❋✐rst ❊❞✐t✐♦♥✳ ❇♦st♦♥✿ ❆❧❧②♥ ❛♥❞ ❇❛❝♦♥✱ ✶✾✻✺✳

❬✹❪ ❊◆●❊▲❑■◆● ❘✳ ●❡♥❡r❛❧ ❚♦♣♦❧♦❣②✳ ❍❡❧❞❡r♠❛♥♥ ❱❡r❧❛❣✱ ✶✾✽✾✳

❬✺❪ ●❘❯❊◆❍❆●❊ ●✳ ■♥✜♥✐t❡ ●❛♠❡s ❛♥❞ ●❡♥❡r❛❧✐③❛t✐♦♥s ♦❢ ❋✐rst✲❈♦✉♥t❛❜❧❡ ❙♣❛❝❡s✳ ●❡♥❡r❛❧ ❚♦♣♦❧♦❣② ❛♥❞ ✐ts ❆♣♣❧✐❝❛t✐♦♥s✱ ✶✾✼✻✳

❬✻❪ ❑❊▲▲❊❨ ❏✳▲✳ ●❡♥❡r❛❧ ❚♦♣♦❧♦❣②✳ ❱❛♥ ◆♦str❛♥❞✱ ✶✾✺✺✳

❬✼❪ ▼❯◆❑❘❊❙ ❏✳ ❚♦♣♦❧♦❣② ❛ ❋✐rst ❈♦✉rs❡✱ Pr❡♥t✐❝❡ ❍❛❧❧✱ ✶✾✼✹✳

❬✽❪ ❚❆▼❆❘■❩ ❆✳ ✲ ❈❆❙❆❘❘❯❇■❆❙ ❋✳ ❊❧❡♠❡♥t♦s ❞❡ ❚♦♣♦❧♦❣✐❛●❡♥❡r❛❧✳ ❈✐❡♥❝✐❛s✱ ❯♥✐✈❡rs✐❞❛❞ ❆✉tó♥♦♠❛ ❞❡ ▼❡①✐❝♦✱ ✷✵✶✺✳

❬✾❪ ❚❊▲●❆❘❙❑❨ ❘✳ ❙♣❛❝❡s ❉❡✜♥❡❞ ❜② ❚♦♣♦❧♦❣✐❝❛❧ ●❛♠❡s ✳❋✉♥❞❛♠❡♥t❛❧ ▼❛t❤❡♠❛t✐❝s✱ ✶✾✼✺✳

❬✶✵❪ ❚❊▲●❆❘❙❑❨ ❘✳ ❚♦♣♦❧♦❣✐❝❛❧ ●❛♠❡s✿ ❖♥ t❤❡ ✺✵✱ ❆♥♥✐✈❡rs❛r② ♦❢t❤❡ ❇❛♥❛❝❤✲▼❛♥③✉r ●❛♠❡✱ ❏♦✉r♥❛❧ ♦❢ ▼❛t❤❡♠❛t✐❝s✱ ✶✾✽✼✳

✺✸