Emanuele Rose Romero de Santana

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❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❛ ❇❛❤✐❛ ✲ ❯❋❇❆

■♥st✐t✉t♦ ❞❡ ▼❛t❡♠át✐❝❛ ✲ ■▼

Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ✲ P●▼❆❚ ❉✐ss❡rt❛çã♦ ❞❡ ▼❡str❛❞♦

❉✉❛❧✐❞❛❞❡ ❜✐t♦♣♦❧ó❣✐❝❛ ♣❛r❛ r❡t✐❝✉❧❛❞♦s

❞✐str✐❜✉t✐✈♦s

❡ á❧❣❡❜r❛s ❞❡ ❍❡②t✐♥❣

❊♠❛♥✉❡❧❡ ❘♦s❡ ❘♦♠❡r♦ ❞❡ ❙❛♥t❛♥❛

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❉✉❛❧✐❞❛❞❡ ❜✐t♦♣♦❧ó❣✐❝❛ ♣❛r❛ r❡t✐❝✉❧❛❞♦s

❞✐str✐❜✉t✐✈♦s

❡ á❧❣❡❜r❛s ❞❡ ❍❡②t✐♥❣

❊♠❛♥✉❡❧❡ ❘♦s❡ ❘♦♠❡r♦ ❞❡ ❙❛♥t❛♥❛

❉✐ss❡rt❛çã♦ ❞❡ ▼❡str❛❞♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ ❈♦❧❡❣✐❛❞♦ ❞❛ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❛ ❇❛❤✐❛ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳

❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ❆♥❞r❡❛s ❇❡r♥❤❛r❞ ▼✐❝❤❛❡❧ ❇r✉♥♥❡r✳

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❙❛♥t❛♥❛✱ ❊♠❛♥✉❡❧❡ ❘♦s❡ ❘♦♠❡r♦ ❞❡✳

❉✉❛❧✐❞❛❞❡ ❜✐t♦♣♦❧ó❣✐❝❛ ♣❛r❛ r❡t✐❝✉❧❛❞♦s ❞✐str✐❜✉t✐✈♦s ❡ á❧❣❡❜r❛s ❞❡ ❍❡②t✐♥❣✳ ✴ ❊♠❛♥✉❡❧❡ ❘♦s❡ ❘♦♠❡r♦ ❞❡ ❙❛♥t❛♥❛✳ ✕ ❙❛❧✈❛❞♦r✿ ❯❋❇❆✱ ✷✵✶✷✳

✼✼ ❢✳ ✿ ✐❧✳

❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ❆♥❞r❡❛s ❇❡r♥❤❛r❞ ▼✐❝❤❛❡❧ ❇r✉♥♥❡r✳

❉✐ss❡rt❛çã♦ ✭♠❡str❛❞♦✮ ✕ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❛ ❇❛❤✐❛✱ ■♥st✐t✉t♦ ❞❡ ▼❛t❡♠át✐❝❛✱ Pr♦❣r❛♠❛ ❞❡ Pós✲❣r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛✱ ✷✵✶✷✳

❘❡❢❡rê♥❝✐❛s ❜✐❜❧✐♦❣rá✜❝❛s✳

✶✳ ❚♦♣♦❧♦❣✐❛ ●❡r❛❧✳ ✷✳ ❚❡♦r✐❛ ❞♦s ❘❡t✐❝✉❧❛❞♦s✳ ✸✳ ❚❡♦r✐❛ ❞❡ ❈❛t❡❣♦r✐❛s✳ ■✳ ❇r✉♥♥❡r✱ ❆♥❞r❡❛s ❇❡r♥❤❛r❞ ▼✐❝❤❛❡❧✳ ■■✳ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❛ ❇❛❤✐❛✱ ■♥st✐✲ t✉t♦ ❞❡ ▼❛t❡♠át✐❝❛✳ ■■■✳ ❚ít✉❧♦✳

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❉✉❛❧✐❞❛❞❡ ❜✐t♦♣♦❧ó❣✐❝❛ ♣❛r❛ r❡t✐❝✉❧❛❞♦s

❞✐str✐❜✉t✐✈♦s ❡ á❧❣❡❜r❛s ❞❡ ❍❡②t✐♥❣✳

❊♠❛♥✉❡❧❡ ❘♦s❡ ❘♦♠❡r♦ ❞❡ ❙❛♥t❛♥❛

❉✐ss❡rt❛çã♦ ❞❡ ▼❡str❛❞♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ ❈♦❧❡❣✐❛❞♦ ❞❛ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❛ ❇❛❤✐❛ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✱ ❛♣r♦✈❛❞❛ ❡♠ ✵✽ ❞❡ ♠❛rç♦ ❞❡ ✷✵✶✷✳

❇❛♥❝❛ ❡①❛♠✐♥❛❞♦r❛✿

Pr♦❢✳ ❉r✳ ❆♥❞r❡❛s ❇❡r♥❤❛r❞ ▼✐❝❤❛❡❧ ❇r✉♥♥❡r✭❖r✐❡♥t❛❞♦r✮ ❯❋❇❆

Pr♦❢✳ ❉r✳ ❍✉❣♦ ▲✉✐③ ▼❛r✐❛♥♦ ❯❙P

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❆❣r❛❞❡❝✐♠❡♥t♦s

✏❊❧❡✈♦ ♠❡✉s ♦❧❤♦s ❛♦s ♠♦♥t❡s✱ ❞❡ ♦♥❞❡ ♠❡ ✈✐rá ♦ s♦❝♦rr♦❄ ❖ ♠❡✉ s♦❝♦rr♦ ✈❡♠ ❞♦ ❙❡♥❤♦r✱ q✉❡ ❢❡③ ♦s ❝é✉s ❡ ❛ t❡rr❛✑✳ ❆❣r❛❞❡ç♦ ❛♦ ♠❡✉ ❉❡✉s✱ ♣♦rq✉❡ ❡♠ t♦❞❛s ❛s ♠✐♥❤❛s ❛✢✐çõ❡s ❊❧❡ ♠❡ ♦✉✈❡ ❡ s♦❝♦rr❡✳ ✏❊ ❡♠ t✐ ❝♦♥✜❛rã♦ ♦s q✉❡ ❝♦♥❤❡❝❡♠ t❡✉ ♥♦♠❡❀ ♣♦rq✉❡ t✉✱ ❙❡♥❤♦r✱ ♥✉♥❝❛ ❞❡s❛♠♣❛r❛st❡ ♦s q✉❡ t❡ ❜✉s❝❛♠✳✑ ❖ ❝❛♠✐♥❤♦ ❞❛ ▼❛t❡♠át✐❝❛ é ✉♠❛ ❞✉❛❧✐❞❛❞❡✿ ❛s ❝♦♥q✉✐st❛s sã♦ ♠✉✐t♦ ♣r❛③❡r♦s❛s ♠❛s t❛♠❜é♠ t❡ ❡①✐❣❡♠ ♠✉✐t♦❀ ❡❧❛s ❡①✐❣❡♠ t❡♠♣♦✱ ❞❡❞✐❝❛çã♦ q✉❛s❡ ❡①❝❧✉s✐✈❛✱ s❛ú❞❡ ✭♦s ❡stô♠❛❣♦s ❞♦s ♠❛t❡♠át✐❝♦s q✉❡ ♦ ❞✐❣❛♠✮✱ ❛❧é♠ ❞❡ ❡①✐❣✐r q✉❡ s❡ t❡♥❤❛ ♦ ❧❛❞♦ ❡♠♦❝✐♦♥❛❧ ❜❛st❛♥t❡ ❡q✉✐❧✐❜r❛❞♦✳ ▼❛s ❊❧❡ s❡♠♣r❡ ♠❡ ♠❛♥t❡✈❡ ✜r♠❡✱ t♦❞❛s ❛s ✈❡③❡s q✉❡ ♣❡♥s❡✐ ❡♠ ❞❡s✐st✐r✳ ✏❉❡✉s é ♦ q✉❡ ♠❡ ❝✐♥❣❡ ❞❡ ❢♦rç❛ ❡ ❛♣❡r❢❡✐ç♦❛ ♦ ♠❡✉ ❝❛♠✐♥❤♦✑✳ ❊ ♦ q✉❡ ❞✐③❡r q✉❛♥❞♦ ❛ ✈✐❞❛✱ ♦✉ ♣✐♦r✱ ❛ ✈✐♦❧ê♥❝✐❛ ❞❡st❡ ♠✉♥❞♦ t❡ r❡s❡r✈❛ s✉r♣r❡s❛s ❡ tr❛✉♠❛s❄ ◆ã♦ ❢♦ss❡ ♣♦r ♠❡✉ ❉❡✉s t❡r ♥♦s ♣r♦t❡❣✐❞♦✱ ♠❡✉ ❡s♣♦s♦✱ ♠✐♥❤❛ ✜❧❤❛ ❡ ❡✉ ♥ã♦ ❡st❛rí❛♠♦s ♠❛✐s ❛q✉✐✱ ❛♣ós ❛q✉❡❧❛ t❡♥t❛t✐✈❛ ❞❡ ❤♦♠✐❝í❞✐♦ tã♦ ❞❡s❡s♣❡r❛❞♦r❛ q✉❡ s♦❢r❡♠♦s✳ ▼❛s ❊❧❡ ♥♦s ❧✐✈r♦✉ ❞❛ ♠♦rt❡✳ ✏❖ ❛♥❥♦ ❞♦ ❙❡♥❤♦r ❛❝❛♠♣❛✲s❡ ❛♦ r❡❞♦r ❞♦s q✉❡ ♦ t❡♠❡♠✱ ❡ ♦s ❧✐✈r❛✳ ▼✐❧ ❝❛✐rã♦ ❛♦ t❡✉ ❧❛❞♦ ❡ ❞❡③ ♠✐❧ à t✉❛ ❞✐r❡✐t❛✱ ♠❛s t✉ ♥ã♦ s❡rás ❛t✐♥❣✐❞♦✳✑ ❊♥✜♠✱ ❡✉ ♥ã♦ s❡✐ ✈✐✈❡r s❡♠ ❡st❡ ❉❡✉s ♠❛r❛✈✐❧❤♦s♦✱ q✉❡ t❡♠ ✉♠ ❛♠♦r ❞❡s♠❡❞✐❞♦ ♣♦r ♠✐♠✱ q✉❡ ❝✉✐❞❛ ❞❡ ♠✐♠ ❡ ❞❡ ♠✐♥❤❛ ❢❛♠í❧✐❛✱ q✉❡ ♥♦s ❣✉❛r❞❛ ❡ ♥♦s ❢♦rt❛❧❡❝❡✳

❆❣r❛❞❡ç♦ ❛♦ ♠❡✉ ❡s♣♦s♦ ❡ ❛♠✐❣♦✱ ❋❡❧✐♣❡ ▼♦s❝♦③♦✱ ♣♦r s❡r ✉♠ ❡①❝❡❧❡♥t❡ ❝♦♠♣❛✲ ♥❤❡✐r♦✱ ✉♠ ♦♠❜r♦ ❛♠✐❣♦ ♣❛r❛ t♦❞♦s ♦s ♠♦♠❡♥t♦s ❡ ✉♠❛ ♣❡ss♦❛ ♠✉✐t♦ ❝❡♥tr❛❞❛✱ q✉❡ ♠❡ ❡♥❝♦r❛❥❛ ❡ ✐♥❝❡♥t✐✈❛ ♠✉✐t♦✳ ❆❣r❛❞❡ç♦ ❛ ❡st❛ ♣❡ss♦❛ tã♦ ❡s♣❡❝✐❛❧ ♣♦r s❡r ♦ ♠❡✉ ❛♠♦r✱ ♦ ♠❡✉ ❛❧✐❝❡r❝❡✱ ♣♦r t❡r ♠❡ ❡s❝♦❧❤✐❞♦ ❝♦♠♦ s✉❛ ✏❛✉①✐❧✐❛❞♦r❛✑ ✭❡s♣♦s❛✮ ❡ ♣♦r s❡r ✉♠ ❡①❝❡❧❡♥t❡ ♣❛✐✳ ❆❣r❛❞❡ç♦ ❛ ♠✐♥❤❛ ✜❧❤❛✱ ❙♦✜❛ ❘♦s❡✱ ♣♦r s❡r ♣❛r❛ ♠✐♠ ❝♦♠♦ ✉♠ ♣❡❞❛❝✐♥❤♦ ❞♦ ❝é✉✱ ✐st♦ é✱ ♣♦r t❡r ❛ ❝❛♣❛❝✐❞❛❞❡ ❞❡ ♠❡ t✐r❛r ❞❡st❡ ♠✉♥❞♦✱ ♠❡ ❢❛③❡r ❡sq✉❡❝❡r ♦s ♣r♦❜❧❡♠❛s ❡ ✈✐✈❡r ♠♦♠❡♥t♦s ❞❡ ♣✉r❛ ❢❡❧✐❝✐❞❛❞❡✳ ❆❣r❛❞❡ç♦ ❛ ❡❧❛ ♣♦r r❡❝❡❜❡r ♠✉✐t♦ ❛♠♦r ❡ ❝❛r✐♥❤♦ ❞❡ ♠✐♠✱ ❡ ♣♦r ❞❡✈♦❧✈ê✲❧♦s ❞❡ ♠❛♥❡✐r❛ tã♦ ❡s♣❡❝✐❛❧✱ t♦❞♦s ♦s ❞✐❛s✳ ❙♦✜❛✱ ❞❡✜♥✐t✐✈❛♠❡♥t❡✱ é ♦ q✉❡ ❡①✐st❡ ❞❡ ♠❛✐s ❧✐♥❞♦ ♥❡st❡ ♣❧❛♥❡t❛✳

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♣❡❧♦ s❡✉ ❛♠♦r ✐♥❝♦♥❞✐❝✐♦♥❛❧ ♣♦r ♠✐♠✳ ❯s❛♥❞♦ ❞♦ ♠❡✉ ❜❛✐❛♥ês q✉❡ ♠✉✐t♦ ❣♦st♦✿ s❡♠ ♠❛✐♥❤❛✱ ❡✉ ♥ã♦ t❡r✐❛ ❝❤❡❣❛❞♦ ❛té ❛q✉✐✳ ❆❣r❛❞❡ç♦ t❛♠❜é♠ ❛♦ ♠❡✉ ♣❛✐✱ ♣♦rq✉❡ s❡♠♣r❡ ♠❡ ✐♥❝❡♥t✐✈♦✉ ❡ ♣♦r t❡r ♠❡ ❢❡✐t♦ ❣♦st❛r ❞❡ ♠❛t❡♠át✐❝❛✳ ❆❣r❛❞❡ç♦ ❛ ❡❧❡ ♣♦r t♦❞♦ ❛♣♦✐♦ ❡ ❛❥✉❞❛ q✉❡ s❡♠♣r❡ ♠❡ ❞❡✉✳ ❆❣r❛❞❡ç♦ ❛♦ ♠❡✉ ❛✈ô✱ s❡✉ ▼❛♥♦❡❧✱ ♣❡❧♦ ❛♣♦✐♦ ❡ ✐♥❝❡♥t✐✈♦ q✉❡ ❡❧❡ s❡♠♣r❡ ❞❡✉✱ t❛♥t♦ ❛ ♠✐♠ q✉❛♥t♦ ❛ ♠✐♥❤❛ ❢❛♠í❧✐❛✳ ❆❣r❛❞❡ç♦ t❛♠❜é♠ ❛♦s ♠❡✉s ✐r♠ã♦s ♣❡❧♦ ❝❛r✐♥❤♦✱ ♣❡❧❛ t♦r❝✐❞❛ ❡ ♣❡❧♦ ❛♣♦✐♦✳ ❆❣r❛❞❡ç♦ às ♠✐♥❤❛s ❝✉♥❤❛❞❛s✱ Pr✐s❝✐❧❧❛ ❡ ❘♦sâ♥❣❡❧❛✱ ♣❡❧♦ ❝❛r✐♥❤♦ ❡ ♣❡❧♦ ❛✉①í❧✐♦✳ ❆❣r❛❞❡ç♦ às ♠✐♥❤❛s ❝♦♠❛❞r❡s✱ Pr✐s❝✐❧❧❛ ❡ ❏✉❧✐❛♥❛✱ ♣❡❧❛ ❛♠✐③❛❞❡ ❡ ❝❛r✐♥❤♦ q✉❡ tê♠ ♣♦r ♠✐♠ ❡ ♣♦r ♠✐♥❤❛ ❢❛♠í❧✐❛✳ ❆❣r❛❞❡ç♦ ❛♦s ♠❡✉s ❧✐♥❞♦s s♦❜r✐♥❤♦s✱ ♣♦r s❡r❡♠ ✉♠❛ ❛❧❡❣r✐❛ ❡♠ ♠✐♥❤❛ ✈✐❞❛ ❡ ♣❡❧♦ ❝❛r✐♥❤♦ q✉❡ ❞♦✉ ❡ r❡❝❡❜♦ ❞❡❧❡s✳

❆❣r❛❞❡ç♦ ❛♦s ♠❡✉s s♦❣r♦s ♣♦r t❡r❡♠ ♠❡ ❛❝♦❧❤✐❞♦ ❡ ❛❥✉❞❛❞♦ ❡♠ ♠✉✐t♦s ♠♦♠❡♥t♦s ✐♠♣♦rt❛♥t❡s ❞❛ ♠✐♥❤❛ ✈✐❞❛✳ ❆❣r❛❞❡ç♦ t❛♠❜é♠ ♣❡❧♦ ❛♣♦✐♦ ❡ ♣❡❧♦ ✐♥❝❡♥t✐✈♦✳ ❆❣r❛❞❡ç♦ ❛♦s ♠❡✉s ❝✉♥❤❛❞♦s✱ ✐r♠ã♦s ❞♦ ♠❡✉ ❡s♣♦s♦✱ ♣♦r t❡r❡♠ ♠❡ ❛❝♦❧❤✐❞♦✱ ❡♠ ❡s♣❡❝✐❛❧ ❛ ♠✐♥❤❛ ❝✉♥❤❛❞❛✱ ❆♠❛♥❞❛✱ ♣♦r s❡✉ ❝❛r✐♥❤♦ ❡ ❛♠✐③❛❞❡✳

❆❣r❛❞❡ç♦ ❛♦ ♠❡✉ ♦r✐❡♥t❛❞♦r✱ ❆♥❞r❡❛s ❇r✉♥♥❡r✱ ♣♦r t♦❞❛ ❛ s✉❛ ❛♠✐③❛❞❡✱ ❝♦♠✲ ♣r❡❡♥sã♦✱ ♣❛❝✐ê♥❝✐❛✱ ❞✐s♣♦s✐çã♦✱ ♣♦r t♦❞♦ ❛♣♦✐♦ ❡ ✐♥❝❡♥t✐✈♦ q✉❡ s❡♠♣r❡ ♠❡ ❞❡✉✱ ♣♦r s❡♠✲ ♣r❡ q✉❡r❡r q✉❡ ❡✉ s✐❣❛ ❛❞✐❛♥t❡ ❡ ♣♦r ♥ã♦ ♠❡ ❞❡✐①❛r ❡s♠♦r❡❝❡r✳ ❆❣r❛❞❡ç♦ ❛ ❡❧❡ ♣♦r t♦❞❛ ♦r✐❡♥t❛çã♦ q✉❡ ♠❡ ❞❡✉✱ t❛♥t♦ ❛❝❛❞ê♠✐❝❛ q✉❛♥t♦ ♥❛ ✈✐❞❛✳ ❆❣r❛❞❡ç♦ ❛ ❆♥❞r❡❛s ♣♦r t❡r ♠❡ ❛♣r❡s❡♥t❛❞♦ ❛ ▲ó❣✐❝❛ ▼❛t❡♠át✐❝❛✱ ❛ ❚❡♦r✐❛ ❞❡ ❘❡t✐❝✉❧❛❞♦s ❡ ❛ ❚❡♦r✐❛ ❞❡ ❈❛t❡❣♦r✐❛s✱ ❡s✲ ♣❡❝✐❛❧♠❡♥t❡ ❡st❛ ú❧t✐♠❛ ♣♦rq✉❡ ♠❡ ♣♦ss✐❜✐❧✐t❛ tr❛❜❛❧❤❛r ❝♦♠ ✈ár✐♦s ♦❜❥❡t♦s ♠❛t❡♠át✐❝♦s✳ ❆❣r❛❞❡ç♦ ❛ ❙❛♠✉❡❧ ❙✐❧✈❛✱ ♠❡✉ ✏❝♦✲♦r✐❡♥t❛❞♦r✑ ♥♦ q✉❡ ❞✐③ r❡s♣❡✐t♦ ❛ ❛❧❣✉♠❛s ❞❡❝✐sõ❡s ❛❝❛❞ê♠✐❝❛s✳ ❙❛♠✉❡❧ s❡ t♦r♥♦✉ ✉♠ ❛♠✐❣♦ ❡ t♦r❝❡❞♦r ❡ s❡♠♣r❡ ♠❡ ❞❡✉ ❞✐❝❛s ❡ ❝♦♥s❡❧❤♦s ✈❛❧♦r♦s♦s✳ ❆❣r❛❞❡ç♦ ❛ ❡❧❡ t❛♠❜é♠ ♣♦r t❡r ♠❡ ❛♣r❡s❡♥t❛❞♦ ❛ ❚♦♣♦❧♦❣✐❛✱ ❝♦♠ ❛ q✉❛❧ ❡✉ ❣♦st♦ ♠✉✐t♦ ❞❡ tr❛❜❛❧❤❛r✳

❆❣r❛❞❡ç♦ ❛♦ Pr♦❢✳ ❍✉❣♦ ▲✉✐③ ▼❛r✐❛♥♦ ❡ à Pr♦❢❛✳ ❱❛❧❡r✐❛ ❱❛③ ❞❡ P❛✐✈❛ ♣♦r ❛❝❡✐t❛r❡♠ ♣❛rt✐❝✐♣❛r ❞❛ ❝♦♠✐ssã♦ ❥✉❧❣❛❞♦r❛ ❞❡ ♠✐♥❤❛ ❞✐ss❡rt❛çã♦ ❡ ♠❡ ❞❛r❡♠ ❛ ❤♦♥r❛ ❞❡ tê✲❧♦s ❝♦♠♦ ♠❡♠❜r♦s ❞❛ ❜❛♥❝❛ ❡①❛♠✐♥❛❞♦r❛ ❞❡ ♠✐♥❤❛ ❞❡❢❡s❛✳

◆ã♦ ♣♦❞❡r✐❛ ❡sq✉❡❝❡r ❞❡ ❛❣r❛❞❡❝❡r ❛♦s ♠❡✉s ❝♦❧❡❣❛s q✉❡r✐❞♦s✱ q✉❡ ❝❛r✐♥❤♦s❛♠❡♥t❡ ♠❡ ❛❞♦t❛r❛♠ ❝♦♠♦ ✏♠❛♠ã❡✑ ❡ ❡✉ ❛ ❡❧❡s ❝♦♠♦ ✏✜❧❤♦s✑ ✕ ❞♦ ♠❛✐s r❡❜❡❧❞❡ à ♠❛✐s ❛♠♦r♦s❛✱ ❡ ♠❡s♠♦ àq✉❡❧❡s q✉❡ ♠❡ ✜③❡r❛♠ s❡♥t✐r ✷✵ ❛♥♦s ♠❛✐s ✈❡❧❤❛✱ ❛♦ ♠❡ ❛✈✐st❛r❡♠ ❞❡ ❧♦♥❣❡ ❡ ❣r✐t❛r❡♠ ✧▼❛♠ã❡✦✧✳ ▼✉✐t♦ ♦❜r✐❣❛❞❛ ♣♦r t♦❞❛ ❛ ❢♦rç❛ ❡ ❝❛r✐♥❤♦ q✉❡ ✈♦❝ês ♠❡ ❞❡r❛♠ ❡♠ ♠✉✐t♦s ♠♦♠❡♥t♦s ❞✐❢í❝❡✐s✳

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♠✐♠✱ ❡s♣❡❝✐❛✐s ❝♦♠♦ ♣r♦❢❡ss♦r❡s✳ ❆❣r❛❞❡ç♦ ❛✐♥❞❛ ❛♦s ♣r♦❢❡ss♦r❡s ❙❛♠✉❡❧ ❙✐❧✈❛ ❡ ❊♥❛❧❞♦ ❱❡r❣❛st❛ ♣♦r t❡r❡♠ ♠❡ ❞❛❞♦ ✉♠ ✈♦t♦ ❞❡ ❝♦♥✜❛♥ç❛ ❡✱ ❝♦♠ ❡❧❡✱ ❛ ♦♣♦rt✉♥✐❞❛❞❡ ❞❡ ❢❛③❡r ❡ ❝♦♥❝❧✉✐r ♦ ♠❡str❛❞♦ ❡♠ ▼❛t❡♠át✐❝❛✳

❋✐♥❛❧♠❡♥t❡✱ ❛❣r❛❞❡ç♦ ❛ ❈❆P❊❙ ♣❡❧♦ ❛♣♦✐♦ ✜♥❛♥❝❡✐r♦ ❝♦♥❝❡❞✐❞♦ ❛ ♠✐♠ ❞✉r❛♥t❡ t♦❞♦ ♦ ♠❡✉ ♠❡str❛❞♦✳

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✏❖ ♠✉♥❞♦ é ✉♠ ❧✉❣❛r ♣❡r✐❣♦s♦ ❞❡ s❡ ✈✐✈❡r✱ ♥ã♦ ♣♦r ❝❛✉s❛ ❞❛q✉❡❧❡s q✉❡ ❢❛③❡♠ ♦ ♠❛❧✱ ♠❛s s✐♠ ♣♦r ❝❛✉s❛ ❞❛q✉❡❧❡s q✉❡ ♦❜s❡r✈❛♠ ❡ ❞❡✐①❛♠ ♦ ♠❛❧ ❛❝♦♥t❡❝❡r✳✑

❆❧❜❡rt ❊✐♥st❡✐♥

✏◆ã♦ ✈♦s ✐♥q✉✐❡t❡✐s ♣♦✐s ♣❡❧♦ ❞✐❛ ❞❡ ❛♠❛♥❤ã✱ ♣♦rq✉❡ ♦ ❞✐❛ ❞❡ ❛♠❛♥❤ã ❝✉✐❞❛rá ❞❡ s✐ ♠❡s♠♦✳ ❇❛st❛ ❛ ❝❛❞❛ ❞✐❛ ♦ s❡✉ ♠❛❧✳✑

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❘❡s✉♠♦

❖ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ t❡♠ ❝♦♠♦ ♦❜❥❡t✐✈♦ ♣r✐♥❝✐♣❛❧ ❛♣r❡s❡♥t❛r ❛ ❡q✉✐✈❛❧ê♥❝✐❛ ❞✉❛❧ ❡♥tr❡ ❛ ❝❛t❡❣♦r✐❛ ❉▲❛t✱ ❞♦s r❡t✐❝✉❧❛❞♦s ❞✐str✐❜✉t✐✈♦s✱ ❡ ❛s ❝❛t❡❣♦r✐❛s Pr✐❡s✱ ❞♦s ❡s♣❛ç♦s Pr✐❡st❧❡②✱ ❡ ❙♣❡❝✱ ❞♦s ❡s♣❛ç♦s ❡s♣❡❝tr❛✐s✱ ♣♦r ♠❡✐♦ ❞♦s ✐s♦♠♦r✜s♠♦s ❡♥tr❡ ❡st❛s ú❧t✐♠❛s ❡ ❛ ❝❛t❡❣♦r✐❛ P❙t♦♥❡✱ ❞♦s ❡s♣❛ç♦s ♣❛✐r✇✐s❡ ❙t♦♥❡❀ t❛♠❜é♠ ❡st❛❜❡❧❡❝❡r ❛ ❡q✉✐✈❛❧ê♥❝✐❛ ❞✉❛❧ ❞❡ ❉▲❛t ❝♦♠ P❙t♦♥❡✱ ♦❜t❡♥❞♦ ❝♦♠ ✐ss♦ ✉♠❛ ❛❧t❡r♥❛t✐✈❛ ♣❛r❛ ♦ ✐s♦♠♦r✜s♠♦ ❞❡ ❈♦r♥✐s❤✳ ❊st❡s ✐s♦♠♦r✜s♠♦s ❡ ❛s ❡q✉✐✈❛❧ê♥❝✐❛s ❞✉❛✐s ❡♥tr❡ ❛s r❡❢❡r✐❞❛s ❝❛t❡❣♦r✐❛s✱ ♣♦ss✐❜✐❧✐t❛r❛♠ ❛s ❞❡s❝r✐çõ❡s ❞✉❛✐s ❞❡ ❛❧❣✉♥s ❝♦♥❝❡✐t♦s ❛❧❣é❜r✐❝♦s ✐♠♣♦rt❛♥t❡s ♥❛ ❚❡♦r✐❛ ❞❡ ❘❡t✐❝✉❧❛❞♦s✱ ❝♦♠♦ ✜❧tr♦s ❡ ✐❞❡❛✐s ❞❡ ✉♠ r❡t✐❝✉❧❛❞♦✳ P♦r ✜♠✱ ❝♦♠♦ ❛❧t❡r♥❛t✐✈❛ ♣❛r❛ ❛ ❞✉❛❧✐❞❛❞❡ ❞❡ ❊s❛❦✐❛✱ sã♦ ❛♣r❡s❡♥t❛❞❛s ❞✉❛❧✐❞❛❞❡s ❜✐t♦♣♦❧ó❣✐❝❛s ♣❛r❛ á❧❣❡❜r❛s ❞❡ ❍❡②t✐♥❣✳

(11)

❆❜str❛❝t

❚❤✐s ✇♦r❦ ❛✐♠s t♦ ♣r❡s❡♥t t❤❡ ❞✉❛❧✐t② ❜❡t✇❡❡♥ t❤❡ ❝❛t❡❣♦r✐❡s ❉▲❛t ♦❢ t❤❡ ❞✐s✲ tr✐❜✉t✐✈❡ ❜♦✉♥❞❡❞ ❧❛tt✐❝❡s✱ ❛♥❞ t❤❡ ❝❛t❡❣♦r✐❡s Pr✐❡s ♦❢ t❤❡ Pr✐❡st❧❡② s♣❛❝❡s ❛♥❞ ❙♣❡❝ ♦❢ t❤❡ s♣❡❝tr❛❧ s♣❛❝❡s✱ ✉s✐♥❣ ✐s♦♠♦r♣❤✐s♠s ❜❡t✇❡❡♥ t❤❡ ❧❛st ♠❡♥t✐♦♥❡❞ ❝❛t❡❣♦r✐❡s ❛♥❞ t❤❡ ❝❛t❡❣♦r② P❙t♦♥❡ ♦❢ t❤❡ ♣❛✐r✇✐s❡ ❙t♦♥❡ s♣❛❝❡s❀ ❛❧s♦ ✇❡ ❡st❛❜❧✐s❤ t❤❡ ❡q✉✐✈❛❧❡♥❝❡ ♦❢ ❉▲❛t ❛♥❞ P❙t♦♥❡✱ r❡❝❡✐✈✐♥❣ s♦ ❛♥ ❛❧t❡r♥❛t✐✈❡ ❢♦r t❤❡ ❈♦r♥✐s❤ ✐s♦♠♦r♣❤✐s♠✳ ❚❤❡s❡ ✐s♦♠♦r✲ ♣❤✐s♠s ❛♥❞ t❤❡ ❡q✉✐✈❛❧❡♥❝✐❡s ❜❡t✇❡❡♥ t❤❡ ❛❜♦✈❡ ♠❡♥t✐♦♥❡❞ ❝❛t❡❣♦r✐❡s✱ ♠❛❦❡ ✐t ♣♦ss✐❜❧❡ t♦ ❣❡t ❞✉❛❧ ❞❡s❝r✐♣t✐♦♥s ♦❢ s♦♠❡ ✐♠♣♦rt❛♥t ❧❛tt✐❝❡ t❤❡♦r❡t✐❝ ♥♦t✐♦♥s✱ ❛s ✜❧t❡rs ❛♥❞ ✐❞❡❛❧s ♦❢ ❛ ❧❛tt✐❝❡✳ ❋✐♥❛❧❧②✱ t♦ ♣r♦♣♦s❡ ❛♥ ❛❧t❡r♥❛t✐✈❡ t♦ t❤❡ ❊s❛❦✐❛ ❞✉❛❧✐t② ❛r❡ s❤♦✇♥ ❜✐t♦♣♦❧♦❣✐❝❛❧ ❞✉❛❧✐t✐❡s ❢♦r ❍❡②t✐♥❣ ❛❧❣❡❜r❛s✳

(12)

❙✉♠ár✐♦

■♥tr♦❞✉çã♦ ✈✐✐

✶ ◆♦çõ❡s ❞❡ ❚♦♣♦❧♦❣✐❛ ●❡r❛❧✱ ❚❡♦r✐❛ ❞❡ ❘❡t✐❝✉❧❛❞♦s ❡ ❈❛t❡❣♦r✐❛s ✶ ✶✳✶ ◆♦çõ❡s ❞❡ ❚♦♣♦❧♦❣✐❛ ●❡r❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✶✳✶✳✶ ❇❛s❡s ❡ s✉❜✲❜❛s❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✶✳✷ ❋❡❝❤♦s ❡ ♣♦♥t♦s ❛❞❡r❡♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✶✳✸ ❋✉♥çõ❡s ❝♦♥tí♥✉❛s ❡ ❤♦♠❡♦♠♦r✜s♠♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✳✶✳✹ ❆①✐♦♠❛s ❞❡ s❡♣❛r❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✶✳✶✳✺ ❈♦♠♣❛❝✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✷ ❖r❞❡♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✷✳✶ ❈♦♥❝❡✐t♦s ❜ás✐❝♦s s♦❜r❡ ♦r❞❡♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✷✳✷ ❈♦♥❥✉♥t♦s ❝r❡s❝❡♥t❡s ❡ ❞❡❝r❡s❝❡♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✶✳✷✳✸ ❋✉♥çõ❡s q✉❡ ♣r❡s❡r✈❛♠ ♦r❞❡♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✶✳✷✳✹ ❈♦t❛s ❡ ❝♦♥❥✉♥t♦s ❞✐r❡❝✐♦♥❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✶✳✸ ❘❡t✐❝✉❧❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✶✳✸✳✶ ❋✐❧tr♦s ❡ ■❞❡❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✶✳✸✳✷ ❘❡t✐❝✉❧❛❞♦s ❞✐str✐❜✉t✐✈♦s ❡s♣❡❝✐❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✶✳✹ ❚❡♦r✐❛ ❞❡ ❈❛t❡❣♦r✐❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✶✳✹✳✶ ◆♦çõ❡s ❜ás✐❝❛s ❞❡ ❈❛t❡❣♦r✐❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✶✳✹✳✷ ❋✉♥t♦r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷

✷ ■s♦♠♦r✜s♠♦s ❡ ❊q✉✐✈❛❧ê♥❝✐❛s ❉✉❛✐s ✷✺

✷✳✶ ❊s♣❛ç♦s ♣❛✐r✇✐s❡ ❙t♦♥❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✷✳✷ ❊s♣❛ç♦s Pr✐❡st❧❡② ❡ ❡s♣❛ç♦s ♣❛✐r✇✐s❡ ❙t♦♥❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✷✳✸ ❊s♣❛ç♦s ♣❛✐r✇✐s❡ ❙t♦♥❡ ❡ ❡s♣❛ç♦s ❡s♣❡❝tr❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✷✳✹ ❘❡t✐❝✉❧❛❞♦s ❞✐str✐❜✉t✐✈♦s ❡ ❡s♣❛ç♦s ♣❛✐r✇✐s❡ ❙t♦♥❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽

✸ ❆❧❣✉♠❛s ❉❡s❝r✐çõ❡s ❉✉❛✐s ❡ ❛ ❉✉❛❧✐❞❛❞❡ ❞❡ ❊s❛❦✐❛ ✺✽ ✸✳✶ ❋✐❧tr♦s ❡ ✐❞❡❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽ ✸✳✶✳✶ ❋✐❧tr♦s ❡ ✐❞❡❛✐s ♣r✐♠♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✺

(13)

✸✳✶✳✷ ❋✐❧tr♦s ❡ ✐❞❡❛✐s ♠❛①✐♠❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✼ ✸✳✷ ■♠❛❣❡♥s ❤♦♠♦♠♦r❢❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✾ ✸✳✸ ❉✉❛❧✐❞❛❞❡s ♣❛r❛ á❧❣❡❜r❛s ❞❡ ❍❡②t✐♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✶

(14)

■♥tr♦❞✉çã♦

❊♠ ♠❡❛❞♦s ❞❡ ✶✾✸✵✱ ▼✳ ❍✳ ❙t♦♥❡ ❞❡✉ ✐♥í❝✐♦ à t❡♦r✐❛ ❞❛ ❞✉❛❧✐❞❛❞❡ ❝♦♠ s❡✉ tr❛✲ ❜❛❧❤♦ ✐♥♦✈❛❞♦r ♥♦ q✉❛❧ ❡❧❡ ❡st❛❜❡❧❡❝❡✉ ✉♠❛ ❡q✉✐✈❛❧ê♥❝✐❛ ❞✉❛❧ ❡♥tr❡ ❛ ❝❛t❡❣♦r✐❛ Bool✱

❞❛s á❧❣❡❜r❛s ❇♦♦❧❡❛♥❛s ❡ ❤♦♠♦♠♦r✜s♠♦s ❞❡ á❧❣❡❜r❛s ❇♦♦❧❡❛♥❛s✱ ❡ ❛ ❝❛t❡❣♦r✐❛ Stone✱

❞♦s ❡s♣❛ç♦s ❝♦♠♣❛❝t♦s ❍❛✉s❞♦r✛ ③❡r♦✲❞✐♠❡♥s✐♦♥❛✐s✱ ❝❤❛♠❛❞♦s ❡s♣❛ç♦s ❙t♦♥❡✱ ❡ ❢✉♥çõ❡s ❝♦♥tí♥✉❛s✳ ❊♠ ✶✾✸✼✱ ♦ ♣ró♣r✐♦ ❙t♦♥❡ ❡st❡♥❞❡✉ ❛ r❡❢❡r✐❞❛ ❞✉❛❧✐❞❛❞❡ ♣❛r❛ ✉♠❛ ❡q✉✐✈❛❧ê♥❝✐❛ ❞✉❛❧ ❡♥tr❡ ❛s ❝❛t❡❣♦r✐❛sDLat✱ ❞♦s r❡t✐❝✉❧❛❞♦s ❞✐str✐❜✉t✐✈♦s ❡ ❧✐♠✐t❛❞♦s ❡ ❤♦♠♦♠♦r✜s♠♦s

❞❡ r❡t✐❝✉❧❛❞♦s ❧✐♠✐t❛❞♦s✱ ❡ Spec✱ ❞♦s ❡s♣❛ç♦s ❡s♣❡❝tr❛✐s ❡ ❢✉♥çõ❡s ❡s♣❡❝tr❛✐s✳

❯♠❛ ❞✉❛❧✐❞❛❞❡ ❡♥tr❡ ❝❛t❡❣♦r✐❛s ❛r❜✐trár✐❛s ♥♦s ❞á✱ ❡♠ ❣❡r❛❧✱ ✉♠❛ ❝❛r❛❝t❡r✐③❛çã♦ ❞❡ ♦❜❥❡t♦s ♠❛t❡♠át✐❝♦s✳ ◆♦ ❝❛s♦ ❞❛ ❞✉❛❧✐❞❛❞❡ ❞❡ ❙t♦♥❡✱ ♦s r❡t✐❝✉❧❛❞♦s ❞✐str✐❜✉t✐✈♦s sã♦ ❝❛r❛❝t❡r✐③❛❞♦s ❛tr❛✈és ❞❡ ❝❡rt♦s ❡s♣❛ç♦s t♦♣♦❧ó❣✐❝♦s✱ ♦s s✉♣r❛❝✐t❛❞♦s ❡s♣❛ç♦s ❡s♣❡❝tr❛✐s✳ ❊st❡s ú❧t✐♠♦s✱ ❡♥tr❡ ♦✉tr❛s ❝♦✐s❛s✱ sã♦ ❡s♣❛ç♦s ❝♦♠♣❛❝t♦sT0 ❡ ❣❡♥❡r❛❧✐③❛♠ ♦s ❡s♣❛ç♦s ❞❡

❙t♦♥❡✳

❊♠ ✶✾✼✵✱ Pr✐❡st❧❡② ❝♦♥str✉✐✉ ✉♠❛ ♦✉tr❛ ❝❛t❡❣♦r✐❛ ❞✉❛❧ à DLat ♣♦r ♠❡✐♦ ❞❡

✏❡s♣❛ç♦s ❙t♦♥❡ ❡s♣❡❝✐❛❧♠❡♥t❡ ♦r❞❡♥❛❞♦s✑ ✕ ❝♦♥❤❡❝✐❞♦s ❝♦♠♦ ❡s♣❛ç♦s Pr✐❡st❧❡② ✕ ♣♦rt❛♥t♦✱ ♠♦str❛♥❞♦ q✉❡ DLat é t❛♠❜é♠ ❞✉❛❧♠❡♥t❡ ❡q✉✐✈❛❧❡♥t❡ à ❝❛t❡❣♦r✐❛ Pries ❞♦s ❡s♣❛ç♦s

Pr✐❡st❧❡② ❡ ❢✉♥çõ❡s ❝♦♥tí♥✉❛s ♣r❡s❡r✈❛♥❞♦ ♦r❞❡♠✳ ❈♦♠♦ r❡s✉❧t❛❞♦✱ t❡♠♦s q✉❡ Pries ❡ Spec sã♦ ❡q✉✐✈❛❧❡♥t❡s✳ ▼❛✐s ❛✐♥❞❛✿ ❡♠ ✶✾✼✺✱ ❈♦r♥✐s❤ ♠♦str♦✉ q✉❡ Spec ❡ Pries sã♦

✐s♦♠♦r❢❛s✳

❉❛❞♦ ✉♠ ❡s♣❛ç♦ Pr✐❡st❧❡② (X, τ,≤)✱ ❡stã♦ ❛ss♦❝✐❛❞❛s ❛ ❡❧❡ ❞✉❛s t♦♣♦❧♦❣✐❛s ♥❛t✉✲

r❛✐s✿ ❛ t♦♣♦❧♦❣✐❛ ❝r❡s❝❡♥t❡✱τ1✱ ❝♦♥s✐st✐♥❞♦ ❞♦s ❛❜❡rt♦s ❝r❡s❝❡♥t❡s ❞❡(X, τ,≤)✱ ❡ ❛ t♦♣♦❧♦❣✐❛

❞❡❝r❡s❝❡♥t❡✱τ2✱ ❝♦♥s✐st✐♥❞♦ ❞♦s ❛❜❡rt♦s ❞❡❝r❡s❝❡♥t❡s ❞❡(X, τ,≤)✳ ❉❡ss❛ ❢♦r♠❛✱(X, τ1, τ2)

é ✉♠ ❡s♣❛ç♦ ❜✐t♦♣♦❧ó❣✐❝♦✳ ❆❞❡♠❛✐s✱ ❛♠❜❛s ❛s t♦♣♦❧♦❣✐❛sτ1 ❡τ2 sã♦ ❡s♣❡❝tr❛✐s✳ ❆ t♦♣♦❧♦✲

❣✐❛ Pr✐❡st❧❡②✱ τ✱ ❞❡ ❢❛t♦✱ é ❣❡r❛❞❛ ♣❡❧❛ ✉♥✐ã♦ ❞❡ τ1 ❡ τ2✱ ❡ ♦ ❡s♣❛ç♦ ❡s♣❡❝tr❛❧ ❛ss♦❝✐❛❞♦

❝♦♠(X, τ,≤)é ♦❜t✐❞♦ ❞❡ (X, τ1, τ2)s✐♠♣❧❡s♠❡♥t❡ ❡sq✉❡❝❡♥❞♦ τ2✳ ❊♥tã♦✱ ❛ r❡♣r❡s❡♥t❛çã♦

❞♦s ❡s♣❛ç♦s Pr✐❡st❧❡② ✕ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱ ❞♦s ❡s♣❛ç♦s ❡s♣❡❝tr❛✐s ✕ ❛tr❛✈és ❞♦s ❡s♣❛ç♦s ❜✐t♦♣♦❧ó❣✐❝♦s é ✉♠❛ q✉❡stã♦ ♥❛t✉r❛❧✳ ❉❡ ❢❛t♦✱ ❏✉♥❣ ❡ ▼♦s❤✐❡r ❞❡♠♦♥str❛r❛♠✱ ❡♠ ✷✵✵✻✱ q✉❡ ♦s r❡t✐❝✉❧❛❞♦s ❞✐str✐❜✉t✐✈♦s ♣♦❞❡♠ s❡r r❡♣r❡s❡♥t❛❞♦s ♣♦r ♠❡✐♦ ❞♦s ❡s♣❛ç♦s ❜✐t♦♣♦❧ó❣✐✲ ❝♦s ✭❝❢✳❬❏▼✵✻❪✮✳ ❊♠ ✷✵✵✽✱ ●✳ ❡ ◆✳ ❇❡③❤❛♥✐s❤✈✐❧✐✱ ❉✳ ●❛❜❡❧❛✐❛✱ ❆✳ ❑✉r③ ✜③❡r❛♠✱ ❡♠ ❬❇❇●❑✶✵❪✱ ✉♠❛ ❛①✐♦♠❛t✐③❛çã♦ ❡①♣❧í❝✐t❛ ❞❛ ❝❧❛ss❡ ❞♦s ❡s♣❛ç♦s ❜✐t♦♣♦❧ó❣✐❝♦s ♦❜t✐❞♦s ❞❛

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♠❛♥❡✐r❛ ❞❡s❝r✐t❛ ❛♥t❡r✐♦r♠❡♥t❡ ✕ ❝❤❛♠❛❞♦s ♣♦r ❡❧❡s ❞❡ ❡s♣❛ç♦s ♣❛✐r✇✐s❡ ❙t♦♥❡ ✕ ❣❡♥❡r❛❧✐✲ ③❛♥❞♦ ❛ ❞❡✜♥✐çã♦ ❞♦s ❡s♣❛ç♦s ❙t♦♥❡ ♣❛r❛ ♦ ❝♦♥t❡①t♦ ❜✐t♦♣♦❧ó❣✐❝♦✳ ❈♦♠♦ ❡s♣❛ç♦s ♣❛✐r✇✐s❡ ❙t♦♥❡ ♥♦s ❢♦r♥❡❝❡ ✉♠❛ ♠❛♥❡✐r❛ ❞❡ s❡ ✏♠♦✈❡r✑ ❞♦s ❡s♣❛ç♦s Pr✐❡st❧❡② ♣❛r❛ ♦s ❡s♣❛ç♦s ❡s♣❡❝✲ tr❛✐s ❡ ✈✐❝❡✲✈❡rs❛✱ ♦ ✐s♦♠♦r✜s♠♦ ❞❡ ❈♦r♥✐s❤ ♣❛r❛Pries❡Spec♣♦❞❡ s❡r ❡st❛❜❡❧❡❝✐❞♦ ♠❛✐s

♥❛t✉r❛❧♠❡♥t❡✱ ♣r✐♠❡✐r♦ ♠♦str❛♥❞♦ q✉❡ Pries é ✐s♦♠♦r❢❛ à PStone ❞♦s ❡s♣❛ç♦s ♣❛✐r✇✐s❡

❙t♦♥❡ ❡ ❢✉♥çõ❡s ❜✐❝♦♥tí♥✉❛s✱ ❞❡♣♦✐s ♠♦str❛♥❞♦ q✉❡ PStone é ✐s♦♠♦r❢❛ à Spec✳ ❆ ❛♣r❡✲

s❡♥t❛çã♦ ❞❡ss❡ ✐s♦♠♦r✜s♠♦ ❛tr❛✈és ❞♦s ❡s♣❛ç♦s ❜✐t♦♣♦❧ó❣✐❝♦s é ♦ ♣r✐♥❝✐♣❛❧ ♦❜❥❡t✐✈♦ ❞❡st❡ tr❛❜❛❧❤♦✳ ❱❛❧❡ r❡ss❛❧t❛r q✉❡ ♦s ❡s♣❛ç♦s ♣❛✐r✇✐s❡ ❙t♦♥❡ ❝❛rr❡❣❛♠ ❛ s✐♠❡tr✐❛ ♣r❡s❡♥t❡ ♥♦s ❡s♣❛ç♦s Pr✐❡st❧❡② ✭❡ r❡t✐❝✉❧❛❞♦s ❞✐str✐❜✉t✐✈♦s✮ ❡ ♦❝✉❧t❛ ♥♦s ❡s♣❛ç♦s ❡s♣❡❝tr❛✐s✳

❆❧é♠ ❞✐ss♦✱ ❛ ♣r♦✈❛ ❞❡ q✉❡DLat é ❞✉❛❧♠❡♥t❡ ❡q✉✐✈❛❧❡♥t❡ à PStone é ♠❛✐s s✐♠♣❧❡s ❞♦

q✉❡ ❛s ♣r♦✈❛s ❡①✐st❡♥t❡s ❞❛ ❡q✉✐✈❛❧ê♥❝✐❛ ❞✉❛❧ ❞❡DLat ❝♦♠ Spec ❡ Pries✳

❖s ❡s♣❛ç♦s Pr✐❡st❧❡② sã♦ ♠❛✐s ✉t✐❧✐③❛❞♦s ♥❛ ❧ó❣✐❝❛✱ ✈✐st♦ q✉❡ ❡❧❡s ♣♦ss✉❡♠ ✉♠❛ ♦r❞❡♠ ❡ ♦r❞❡♥s sã♦ ✐♠♣♦rt❛♥t❡s ♣❛r❛ ❛ ❧ó❣✐❝❛✳ ❉❡ss❛ ❢♦r♠❛✱ ❛ ❞✉❛❧✐❞❛❞❡ ❞❡ Pr✐❡st❧❡② s❡ t♦r♥♦✉ ❝♦♠✉♠ ❡♥tr❡ ❧ó❣✐❝♦s✳ ❊♥tã♦✱ ♠❛✐s ❞✉❛❧✐❞❛❞❡s ♣❛r❛ r❡t✐❝✉❧❛❞♦s ❞✐str✐❜✉t✐✈♦s tê♠ s✐❞♦ r❡❛❧✐③❛❞❛s ❡♠ t❡r♠♦s ❞❡ ❡s♣❛ç♦s Pr✐❡st❧❡②✱ ❝♦♠♦✱ ♣♦r ❡①❡♠♣❧♦✱ ❛ ❞✉❛❧✐❞❛❞❡ ❞❡ ❊s❛❦✐❛✳ ◆❡st❡ t❡①t♦✱ ♠❡♥❝✐♦♥❛r❡♠♦s ❛♣❡♥❛s ✉♠❛ ❛❧t❡r♥❛t✐✈❛ ♣❛r❛ ❛ ❞✉❛❧✐❞❛❞❡ ❞❡ ❊s❛❦✐❛ ♣❛r❛ á❧❣❡❜r❛s ❞❡ ❍❡②t✐♥❣✱ ✐♥tr♦❞✉③✐♥❞♦ ❛s s✉❜❝❛t❡❣♦r✐❛s ❞❡ PStone ❡ Spec✱ q✉❡ sã♦

✐s♦♠♦r❢❛s à Esa ❞♦s ❡s♣❛ç♦s ❞❡ ❊s❛❦✐❛ ❡ ❞✉❛❧♠❡♥t❡ ❡q✉✐✈❛❧❡♥t❡s à ❝❛t❡❣♦r✐❛ Heyt ❞❛s

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❈❛♣ít✉❧♦ ✶

◆♦çõ❡s ❞❡ ❚♦♣♦❧♦❣✐❛ ●❡r❛❧✱ ❚❡♦r✐❛ ❞❡

❘❡t✐❝✉❧❛❞♦s ❡ ❈❛t❡❣♦r✐❛s

◆♦ ♣r❡s❡♥t❡ ❝❛♣ít✉❧♦✱ s❡rã♦ ❛♣r❡s❡♥t❛❞♦s r❡s✉❧t❛❞♦s ❡ ❞❡✜♥✐çõ❡s r❡❧❡✈❛♥t❡s ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡st❡ tr❛❜❛❧❤♦✳ P♦r s❡ tr❛t❛r ❞❛ ❝♦♥str✉çã♦ ❞❡ ❞✉❛❧✐❞❛❞❡s ❝❛t❡❣♦r✐❛✐s ❡♥tr❡ r❡t✐❝✉❧❛❞♦s ❞✐str✐❜✉t✐✈♦s ❡ ❡s♣❛ç♦s ❜✐t♦♣♦❧ó❣✐❝♦s ❡s♣❡❝✐❛✐s✱ ❜❡♠ ❝♦♠♦ ❞❛ ❝♦♥str✉çã♦ ❞♦s ✐s♦♠♦r✜s♠♦s ❡♥tr❡ ❡st❡s ❡s♣❛ç♦s ❜✐t♦♣♦❧ó❣✐❝♦s✱ ❡s♣❛ç♦s t♦♣♦❧ó❣✐❝♦s ❡s♣❡❝✐❛❧♠❡♥t❡ ♦r✲ ❞❡♥❛❞♦s ❡ ❡s♣❛ç♦s ❡s♣❡❝tr❛✐s✱ é ♣r❡❝✐s♦ t❡r ❡♠ ♠❡♥t❡ ❝♦♥❝❡✐t♦s t♦♣♦❧ó❣✐❝♦s ❜ás✐❝♦s✱ ❜❡♠ ❝♦♠♦ ❡①♣♦r ♣❛rt❡ ❞❛ t❡♦r✐❛ ❞❛s ♦r❞❡♥s ❡ ❞❡ r❡t✐❝✉❧❛❞♦s✱ ❡♥❣❧♦❜❛♥❞♦ ♦s ❝♦♥❝❡✐t♦s ❞❡ r❡t✐✲ ❝✉❧❛❞♦s ❞✐str✐❜✉t✐✈♦s✱ ✐♥❝❧✉s✐✈❡ ❛s á❧❣❡❜r❛s ❞❡ ❍❡②t✐♥❣✳ P♦r ✜♠✱ s❡rã♦ ❡s❝❧❛r❡❝✐❞❛s ♥♦çõ❡s ❜ás✐❝❛s ❝❛t❡❣♦r✐❛✐s✳

❆♦ ❧♦♥❣♦ ❞♦ t❡①t♦✱ P(X) s❡rá ✉t✐❧✐③❛❞♦ ♣❛r❛ ❞❡♥♦t❛r ♦ ❝♦♥❥✉♥t♦ ❞❛s ♣❛rt❡s ❞❡

✉♠ ❞❛❞♦ ❝♦♥❥✉♥t♦ X ❡ Ac✱ ❢r❡q✉❡♥t❡♠❡♥t❡✱ ❞❡♥♦t❛rá ♦ ❝♦♠♣❧❡♠❡♥t❛r ❞♦ ❝♦♥❥✉♥t♦ A ❚❛♠❜é♠✱C ⊆f B s✐❣♥✐✜❝❛rá q✉❡ C é ✉♠ s✉❜❝♦♥❥✉♥t♦ ✜♥✐t♦ ❞❡ B✳

✶✳✶ ◆♦çõ❡s ❞❡ ❚♦♣♦❧♦❣✐❛ ●❡r❛❧

❊s♣❛ç♦s t♦♣♦❧ó❣✐❝♦s ❡stã♦ ♣r❡s❡♥t❡s ❡♠ q✉❛s❡ t♦❞♦s ♦s r❛♠♦s ❞❛ ♠❛t❡♠át✐❝❛✱ s❡♥❞♦ ♣♦rt❛♥t♦ ✉♠❛ ♣♦♥t❡ ❡♥tr❡ ❞✐✈❡rs❛s t❡♦r✐❛s ♠❛t❡♠át✐❝❛s✱ ❝♦♠♦ s❡rá ✈✐st♦ ❛❞✐❛♥t❡✳ ◆❡st❛ s❡çã♦ sã♦ ❞❡s❝r✐t♦s ❛♣❡♥❛s ♦s ❝♦♥❝❡✐t♦s ❡ ♣r♦♣r✐❡❞❛❞❡s✱ ❛❝❡r❝❛ ❞❡ ❡s♣❛ç♦s t♦♣♦❧ó❣✐✲ ❝♦s✱ q✉❡ s❡rã♦ ✉t✐❧✐③❛❞♦s ♥❛s ❞❡♠♦♥str❛çõ❡s ♣r❡s❡♥t❡s ♥❡st❡ tr❛❜❛❧❤♦ ❡ ♥♦ ❡♥t❡♥❞✐♠❡♥t♦ ❞♦ ♠❡s♠♦✱ t❛✐s ❝♦♠♦ ❜❛s❡s ❡ s✉❜✲❜❛s❡s✱ ❢❡❝❤♦s ❡ ❛❞❡rê♥❝✐❛✱ ❝♦♥t✐♥✉✐❞❛❞❡ ❡ ❤♦♠❡♦♠♦r✜s♠♦s✱ s❡♣❛r❛çã♦ ❡ ❝♦♠♣❛❝✐❞❛❞❡✳ ❆s ❞❡♠♦♥str❛çõ❡s ❞❡st❛s ♣r♦♣r✐❡❞❛❞❡s ♥ã♦ sã♦ ♦ ❢♦❝♦ ❞❡st❡ tr❛❜❛❧❤♦✱ ♣♦rt❛♥t♦ s❡rã♦ ♦♠✐t✐❞❛s✳ ❖ ❧❡✐t♦r ♣♦❞❡rá ❡♥❝♦♥trá✲❧❛s ❡♠ ❬❆❏P❚❪✳

(17)

❉❡✜♥✐çã♦ ✶✳✶✳✶✳ ❯♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ é ✉♠ ♣❛r (X, τ)✱ ♦♥❞❡ X é ✉♠ ❝♦♥❥✉♥t♦ ❡ τ

é ✉♠❛ ❝♦❧❡çã♦ ❞❡ s✉❜❝♦♥❥✉♥t♦s ❞❡ X s❛t✐s❢❛③❡♥❞♦ ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿

(i) ∅ ❡ X ♣❡rt❡♥❝❡♠ ❛ τ✳

(ii) ❆ ✐♥t❡rs❡çã♦ ✜♥✐t❛ ✭♥ã♦✲✈❛③✐❛✮ ❞❡ ❡❧❡♠❡♥t♦s ❞❡ τ é ✉♠ ❡❧❡♠❡♥t♦ ❞❡ τ✳

(iii) ❆ ✉♥✐ã♦ q✉❛❧q✉❡r ❞❡ ❡❧❡♠❡♥t♦s ❞❡ τ é ✉♠ ❡❧❡♠❡♥t♦ ❞❡ τ✳

◆❡st❡ ❝❛s♦✱ ❞✐③✲s❡ q✉❡ τ é ✉♠❛ t♦♣♦❧♦❣✐❛ s♦❜r❡ X ✭♦✉ q✉❡ X ❡stá ♠✉♥✐❞♦ ❞❛

t♦♣♦❧♦❣✐❛ τ✮✳ ❖s ❡❧❡♠❡♥t♦s ❞❡ X sã♦ ❝❤❛♠❛❞♦s ❞❡ ♣♦♥t♦s ❞♦ ❡s♣❛ç♦✳ ❖s ❡❧❡♠❡♥t♦s ❞❡ τ sã♦ ❝❤❛♠❛❞♦s ❞❡ ❛❜❡rt♦s ❞❡ X✳ ◗✉❛♥❞♦ ❡st✐✈❡r ❝❧❛r♦✱ ♥♦ ❝♦♥t❡①t♦✱ q✉❛❧ é ❛ t♦♣♦❧♦❣✐❛

❛ss♦❝✐❛❞❛ ❛♦ ❝♦♥❥✉♥t♦X ❡♠ q✉❡stã♦✱ ❞❡♥♦t❛r✲s❡✲á ♦ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ ❛♣❡♥❛s ♣♦rX✳

❈♦♠♦ ❝♦♥s❡q✉ê♥❝✐❛ ❞❛ ❞❡✜♥✐çã♦ ❞❡ t♦♣♦❧♦❣✐❛ t❡♠✲s❡ q✉❡ ♦ ❝♦♥❥✉♥t♦ ✈❛③✐♦ ❡ ♦ ❡s♣❛ç♦ t♦❞♦ sã♦ ❝♦♥❥✉♥t♦s ❛❜❡rt♦s✱ ❛ ✐♥t❡rs❡çã♦ ✜♥✐t❛ ❞❡ ❛❜❡rt♦s é ✉♠ ❛❜❡rt♦✱ ❜❡♠ ❝♦♠♦ ❛ ✉♥✐ã♦ q✉❛❧q✉❡r ❞❡ ❛❜❡rt♦s✳

❉❡✜♥✐çã♦ ✶✳✶✳✷✳ ❙❡❥❛ (X, τ) ✉♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦✳ ❯♠ s✉❜❝♦♥❥✉♥t♦ F ❞❡ X é ❞✐t♦

❢❡❝❤❛❞♦ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ X\F é ✉♠ ❝♦♥❥✉♥t♦ ❛❜❡rt♦✳

❉❡❝♦rr❡ ❞❛s ❧❡✐s ❞❡ ❉❡ ▼♦r❣❛♥ q✉❡ ♦ ❡s♣❛ç♦ t♦❞♦ ❡ ♦ ❝♦♥❥✉♥t♦ ✈❛③✐♦ sã♦ ❢❡❝❤❛❞♦s✱ ❛ ✉♥✐ã♦ ✜♥✐t❛ ❞❡ ❢❡❝❤❛❞♦s é ✉♠ ❝♦♥❥✉♥t♦ ❢❡❝❤❛❞♦✱ ❜❡♠ ❝♦♠♦ ❛ ✐♥t❡rs❡çã♦ q✉❛❧q✉❡r ❞❡ ❢❡❝❤❛❞♦s✳

❉❡✜♥✐çã♦ ✶✳✶✳✸✳ ❯♠ s✉❜❝♦♥❥✉♥t♦ A ❞❡ ✉♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ X é ❞✐t♦ ❛❜❡rt♦✲❢❡❝❤❛❞♦

s❡✱ ❡ s♦♠❡♥t❡ s❡✱ A ❡ Ac sã♦ ❛❜❡rt♦s ❞❡ X

◆❡st❡ ❝❛s♦✱ A ❡ Ac sã♦ ❢❡❝❤❛❞♦s ❞❡ X✳ ❆❧é♠ ❞✐ss♦✱ ✉♥✐ã♦ ✜♥✐t❛ ❡ ✐♥t❡rs❡çã♦ ✜♥✐t❛ ❞❡ ❛❜❡rt♦s✲❢❡❝❤❛❞♦s é ✉♠ ❝♦♥❥✉♥t♦ ❛❜❡rt♦✲❢❡❝❤❛❞♦✳

❊①❡♠♣❧♦ ✶✳✶✳✹✳ ❖s ❡①❡♠♣❧♦s ♠❛✐s s✐♠♣❧❡s ❞❡ t♦♣♦❧♦❣✐❛s sã♦✿

(1) ❚♦♣♦❧♦❣✐❛ ❞✐s❝r❡t❛✳ ❊st❛ t♦♣♦❧♦❣✐❛ é ❢♦r♠❛❞❛ ♣❡❧❛ ❝♦❧❡çã♦ ❞❡ t♦❞♦s ♦s s✉❜❝♦♥❥✉♥t♦s

❞❡ ✉♠ ❞❛❞♦ ❝♦♥❥✉♥t♦ X✱ ✐✳❡✳✱ τ := P(X)✳ ❉✐③✲s❡ ❡♥tã♦ q✉❡ X ❡stá ♠✉♥✐❞♦ ❞❛ t♦♣♦❧♦❣✐❛

❞✐s❝r❡t❛ ♦✉ q✉❡ (X, τ) é ✉♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ ❞✐s❝r❡t♦✳

(2) ❚♦♣♦❧♦❣✐❛ ❝❛ót✐❝❛✳ ❊st❛ t♦♣♦❧♦❣✐❛ é ❝♦♠♣♦st❛ ❛♣❡♥❛s ♣❡❧♦ ❝♦♥❥✉♥t♦ ✈❛③✐♦ ❡ ♣❡❧♦

❡s♣❛ç♦ t♦❞♦✱ ✐✳❡✳✱ ❞❛❞♦ ✉♠ ❝♦♥❥✉♥t♦X✱ τ :={∅, X}✳

(3) ❯♠ ❡①❡♠♣❧♦ ❝♦♥❤❡❝✐❞♦ ❞❡ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ é ❛ r❡t❛ r❡❛❧ ♠✉♥✐❞❛ ❞❛ t♦♣♦❧♦❣✐❛

✉s✉❛❧✳ ❯♠ s✉❜❝♦♥❥✉♥t♦A ❞❡st❡ ❡s♣❛ç♦ s❡rá ❛❜❡rt♦ ♥❡st❛ t♦♣♦❧♦❣✐❛ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ♣❛r❛

❝❛❞❛ ♣♦♥t♦ x ❞❡ A ❡①✐st❡ ✉♠ ε > 0 t❛❧ q✉❡ ♦ ✐♥t❡r✈❛❧♦ ❛❜❡rt♦ ]x−ε, x+ε[ ❡stá ❝♦♥t✐❞♦

❡♠ A✳

◆ã♦ é ❞✐❢í❝✐❧ ✈❡r✐✜❝❛r q✉❡✱ s❡(X, τ)é ✉♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ ❡Y é ✉♠ s✉❜❝♦♥❥✉♥t♦

(18)

❉❡✜♥✐çã♦ ✶✳✶✳✺✳ ❙❡❥❛ (X, τ) ✉♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ ❡ Y ✉♠ s✉❜❝♦♥❥✉♥t♦ ❞❡ X✳ ❙❡ O =

{Y ∩U|U ∈Y}✱ ❞✐③✲s❡ ❡♥tã♦ q✉❡ (Y,O) é ✉♠ s✉❜❡s♣❛ç♦ ❞❡ ❳ ❡ q✉❡ O é ❛ t♦♣♦❧♦❣✐❛ ✐♥❞✉③✐❞❛ ♣♦r ❳✭ ♦✉ q✉❡ O é ❛ t♦♣♦❧♦❣✐❛ ❞❡ s✉❜❡s♣❛ç♦✮✳

✶✳✶✳✶ ❇❛s❡s ❡ s✉❜✲❜❛s❡s

❉❡✜♥✐çã♦ ✶✳✶✳✻✳ ❙❡❥❛ (X, τ) ✉♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦✳ ❯♠❛ ❝♦❧❡çã♦ B ⊆ τ é ✉♠❛ ❜❛s❡

♣❛r❛ ❡st❡ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ t♦❞♦ ❛❜❡rt♦ ❞❡ X ♣♦❞❡ s❡r ❡s❝r✐t♦ ❝♦♠♦ ❛

✉♥✐ã♦ ❞❡ ✉♠❛ s✉❜❝♦❧❡çã♦ ❞❡ ❡❧❡♠❡♥t♦s ❞❡B✱ ✐✳❡✳✱ ♣❛r❛ t♦❞♦ V ∈τ✱ ❡①✐st❡ B′ ⊆B t❛❧ q✉❡

V =S

B′✳

❖ ♣ró①✐♠♦ r❡s✉❧t❛❞♦ ❝♦♥s✐st❡ ❞❡ ✉♠ ♠ét♦❞♦ ♣❛r❛ s❛❜❡r q✉❛♥❞♦ ✉♠❛ ❞❛❞❛ ❝♦❧❡çã♦ ❞❡ ❛❜❡rt♦s ❞❡ ✉♠❛ t♦♣♦❧♦❣✐❛ é ✉♠❛ ❜❛s❡ ♣❛r❛ ♦ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ ❡♠ q✉❡stã♦✳

Pr♦♣♦s✐çã♦ ✶✳✶✳✼✳ ❙❡❥❛♠ (X, τ) ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ ❡ B ✉♠❛ ❝♦❧❡çã♦ ❞❡ ❛❜❡rt♦s ❞❡ X✳

❊♥tã♦ B é ✉♠❛ ❜❛s❡ ❞❛ t♦♣♦❧♦❣✐❛ τ s❡ ❡ s♦♠❡♥t❡ s❡✱ ♣❛r❛ t♦❞♦ ❛❜❡rt♦ V ❡ t♦❞♦ ♣♦♥t♦ x

❞❡ V✱ ❡①✐st❡ U ∈B t❛❧ q✉❡ x∈U ⊆V✳

❉❡♠♦♥str❛çã♦✿ ❙❡❥❛ B✉♠❛ ❝♦❧❡çã♦ ❞❡ ❛❜❡rt♦s ❞❡ X✳ ❙✉♣♦♥❤❛ q✉❡ B é ✉♠❛ ❜❛s❡ ❞❛

t♦♣♦❧♦❣✐❛ ❞❡ X✳ ❊♥tã♦✱ ♣❛r❛ t♦❞♦ ❛❜❡rt♦ V ❞❡ X✱ ❡①✐st❡ B′ ⊆ B t❛❧ q✉❡ V = [

B∈B′

B✳

▲♦❣♦✱ ♣❛r❛ t♦❞♦ x ∈ V✱ ❡①✐st❡ U ∈ B′ ⊆ B t❛❧ q✉❡ x ∈ U ⊆ [ B∈B′

B✱ ✐st♦ é✱ ♣❛r❛

t♦❞♦ x ∈ V✱ ❡①✐st❡ U ∈ B t❛❧ q✉❡ x ∈ U ⊆ V✳ ❘❡❝✐♣r♦❝❛♠❡♥t❡✱ s❡❥❛ U ⊆ X ❛❜❡rt♦✳

P♦r ❤✐♣ót❡s❡✱ ♣❛r❛ t♦❞♦ x ∈ U ❡①✐st❡ Bx ∈ B t❛❧ q✉❡ x ∈ Bx ⊆ U✳ ❊♥tã♦✱ ❢❛③❡♥❞♦

B′ :={Bx|x∈U}✱ t❡♠✲s❡ q✉❡ B′ ⊆B ❡U =S

x∈UBx = S

B′✳

❈♦♥❢♦r♠❡ ❛ ♣r♦♣♦s✐çã♦ ❛❝✐♠❛✱ ♦s ✐♥t❡r✈❛❧♦s ❛❜❡rt♦s ❞❡s❝r✐t♦s ♥♦ ❊①❡♠♣❧♦ ✶✳✶✳✹(3)

❢♦r♠❛♠ ✉♠❛ ❜❛s❡ ♣❛r❛ ❛ t♦♣♦❧♦❣✐❛ ✉s✉❛❧ ❞❛ r❡t❛✳

❉❡✜♥✐çã♦ ✶✳✶✳✽✳ ❙❡❥❛ B ✉♠❛ ❝♦❧❡çã♦ ❞❡ s✉❜❝♦♥❥✉♥t♦s ❞❡ ✉♠❛ ❝♦♥❥✉♥t♦ X ❡ s❡❥❛ B∗

❛ ❝♦❧❡çã♦ ❞❡ t♦❞♦s ♦s s✉❜❝♦♥❥✉♥t♦s ❞❡ X q✉❡ sã♦ ✉♥✐õ❡s ❞❡ ❡❧❡♠❡♥t♦s ❞❡ B ✭✐♥❝❧✉s✐✈❡ ❛

✉♥✐ã♦ ✈❛③✐❛✮✳ ❙❡B∗ é ✉♠❛ t♦♣♦❧♦❣✐❛ s♦❜r❡X✱ ❡♥tã♦B∗ é ❝❤❛♠❛❞❛ ❞❡ t♦♣♦❧♦❣✐❛ ❣❡r❛❞❛

♣♦r B✱ ❡ B é ✉♠❛ ❜❛s❡ ♣❛r❛ ❛ t♦♣♦❧♦❣✐❛ B∗✳

◆❡♠ s❡♠♣r❡ ✉♠❛ ❝♦❧❡çã♦ ❞❡ s✉❜❝♦♥❥✉♥t♦s ❞❡ ✉♠ ❞❛❞♦ ❝♦♥❥✉♥t♦ X ❣❡r❛ ✉♠❛

t♦♣♦❧♦❣✐❛✳ ❯♠ ❡①❡♠♣❧♦ ❞✐st♦ é ❛ ❝♦❧❡çã♦ B ❞❡ t♦❞♦s ♦s ✐♥t❡r✈❛❧♦s ❢❡❝❤❛❞♦s [a, b]✱ ❝♦♠

a < b✱ ❞❛ r❡t❛ r❡❛❧✳ ❉❡ ❢❛t♦✱ ❛ Pr♦♣♦s✐çã♦ ✶✳✶✳✼ ♥♦s ❣❛r❛♥t❡ q✉❡ B∗ ♥ã♦ é ✉♠❛ t♦♣♦❧♦❣✐❛

s♦❜r❡ R✳ P❛r❛ ✈❡r ✐st♦✱ s✉♣♦♥❤❛ q✉❡ B∗ s❡❥❛ ✉♠❛ t♦♣♦❧♦❣✐❛ s♦❜r❡ R ❡ ✜①❡z ∈R✳ ❊♥tã♦

[z, z+ 1] ❡ [z −1, z] sã♦ ❛❜❡rt♦s ❞❡ B∗✳ ▲♦❣♦✱ [z−1, z]∩[z, z + 1] ={z} é ❛❜❡rt♦✱ ♣♦✐s é ✐♥t❡rs❡çã♦ ✜♥✐t❛ ❞❡ ❛❜❡rt♦s✳ P♦ré♠✱ ♣❛r❛V :={z} ❡ z ∈V ♥ã♦ ❡①✐st❡ ✉♠ ❡❧❡♠❡♥t♦ ❞❡

(19)

Pr♦♣♦s✐çã♦ ✶✳✶✳✾✳ ❙❡❥❛♠ X ✉♠ ❝♦♥❥✉♥t♦ ❡B ✉♠❛ ❝♦❧❡çã♦ ❞❡ s✉❜❝♦♥❥✉♥t♦s ❞❡ X s❛t✐s✲

❢❛③❡♥❞♦ ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿

(1) P❛r❛ ❝❛❞❛ V1, V2 ∈B ❡ ❝❛❞❛ x∈V1∩V2✱ ❡①✐st❡ V ∈B t❛❧ q✉❡ x∈V ⊆V1∩V2✳

(2) P❛r❛ ❝❛❞❛ x∈X✱ ❡①✐st❡ U ∈B t❛❧ q✉❡ x∈U✳

❊♥tã♦✱ ❛ ❝♦❧❡çã♦ B∗✱ ❢♦r♠❛❞❛ ♣❡❧❛s ✉♥✐õ❡s ❞❡ s✉❜❝♦❧❡çõ❡s ❞❡ B é ✉♠❛ t♦♣♦❧♦❣✐❛

s♦❜r❡ X ❡ B é ✉♠❛ ❜❛s❡ ♣❛r❛ ♦ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ (X,B∗)✳

Pr♦♣♦s✐çã♦ ✶✳✶✳✶✵✳ ❙❡❥❛♠X ✉♠ ❝♦♥❥✉♥t♦ ❡C ✉♠❛ ❝♦❧❡çã♦ ❞❡ s✉❜❝♦♥❥✉♥t♦s ❞❡X✳ ❊♥tã♦

T :=T

{τ :τ é t♦♣♦❧♦❣✐❛ s♦❜r❡ X ❡ C ⊆τ} é ✉♠❛ t♦♣♦❧♦❣✐❛ s♦❜r❡ X✳

❉❡♠♦♥str❛çã♦✿

(i) ➱ ❝❧❛r♦ q✉❡ ∅, X ∈ T✳

(ii) ❙❡❥❛ V ⊆ T ✉♠❛ ❝♦❧❡çã♦ q✉❛❧q✉❡r✳ ❱❡❥❛♠♦s q✉❡ S

V ∈ T✳ ❚❡♠♦s q✉❡ V ⊆τ✱ ♣❛r❛

t♦❞♦ τ t♦♣♦❧♦❣✐❛ q✉❡ ❝♦♥té♠ C✳ ❈♦♠♦ τ é t♦♣♦❧♦❣✐❛ ❡ V ⊆ τ✱ t❡♠♦s q✉❡ S

V ∈τ✱

♣❛r❛ t♦❞♦ τ t♦♣♦❧♦❣✐❛ ❝♦♥t❡♥❞♦ C✳ ▲♦❣♦✱ S

V ∈ T

{τ : τ é t♦♣♦❧♦❣✐❛ s♦❜r❡ X ❡

C ⊆ τ}=T✳

(iii) ❙❡❥❛ V ⊆ T ✉♠❛ ❝♦❧❡çã♦ ✜♥✐t❛ q✉❛❧q✉❡r✳ P❛r❛ ✈❡r✐✜❝❛r q✉❡ T

V ♣❡rt❡♥❝❡ ❛ T✱ ❛♣❧✐❝❛✲s❡ ✉♠ r❛❝✐♦❝í♥✐♦ ❛♥á❧♦❣♦ ❛♦ ✉t✐❧✐③❛❞♦ ❡♠ (ii)✳

❉❡✜♥✐çã♦ ✶✳✶✳✶✶✳ ❙❡❥❛ X ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ ❡ B⊆τ✳ ❉✐③❡♠♦s q✉❡ B é ✉♠❛ s✉❜✲❜❛s❡

❞❡ X s❡ ♦ ❝♦♥❥✉♥t♦ ❞❛s ✐♥t❡rs❡çõ❡s ✜♥✐t❛s ❞❡ ❡❧❡♠❡♥t♦s ❞❡ B ❢♦r♠❛ ✉♠❛ ❜❛s❡ ♣❛r❛ X✳

❉❡✜♥✐çã♦ ✶✳✶✳✶✷✳ ❯♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ X é ❞✐t♦ ③❡r♦✲❞✐♠❡♥s✐♦♥❛❧ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ X ♣♦ss✉✐ ✉♠❛ ❜❛s❡ ❞❡ ❝♦♥❥✉♥t♦s ❛❜❡rt♦s✲❢❡❝❤❛❞♦s✳

P❛r❛ ✉♠ ❝♦♥❥✉♥t♦X✱ ♦ ❡①❡♠♣❧♦ ♠❛✐s s✐♠♣❧❡s ❞❡ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ ③❡r♦✲❞✐♠❡♥s✐♦♥❛❧

é(X,P(X))✱ ♦✉ s❡❥❛✱ X ❝♦♠ ❛ t♦♣♦❧♦❣✐❛ ❞✐s❝r❡t❛✳

✶✳✶✳✷ ❋❡❝❤♦s ❡ ♣♦♥t♦s ❛❞❡r❡♥t❡s

❉❡✜♥✐çã♦ ✶✳✶✳✶✸✳ ❙❡❥❛ (X, τ) ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦✳ ❯♠ ♣♦♥t♦ x ∈ A é ❞✐t♦ ✉♠ ♣♦♥t♦

✐♥t❡r✐♦r ❞❡ A s❡ ❡①✐st❡ V ∈ τ t❛❧ q✉❡ x ∈ V ⊆ A✳ ❆♦ ❝♦♥❥✉♥t♦ ❞♦s ♣♦♥t♦s ✐♥t❡r✐♦r❡s

❝❤❛♠❛✲s❡ ✐♥t❡r✐♦r ❞❡ A ❡ ❞❡♥♦t❛✲s❡ ♣♦r Int(A)✳

◆ã♦ é ❞✐❢í❝✐❧ ✈❡r✐✜❝❛r q✉❡ ✉♠ s✉❜❝♦♥❥✉♥t♦ A❞❡ ✉♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦X é ❛❜❡rt♦

s❡✱ ❡ s♦♠❡♥t❡ s❡✱ t♦❞♦s ♦s s❡✉s ♣♦♥t♦s sã♦ ✐♥t❡r✐♦r❡s✱ ✐st♦ é✱ A=Int(A)✳

❉❡✜♥✐çã♦ ✶✳✶✳✶✹✳ ❙❡❥❛♠(X, τ)❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ ❡ A⊆X✳ ❊♥tã♦ ❞❡✜♥❡✲s❡ ♦ ❢❡❝❤♦ ❞♦

❝♦♥❥✉♥t♦ A ❝♦♠♦ ❛ ✐♥t❡rs❡çã♦ ❞❡ t♦❞♦s ♦s ❢❡❝❤❛❞♦s q✉❡ ❝♦♥té♠ A✳ ❖ ❢❡❝❤♦ ❞❡ A s❡rá

(20)

❈♦♥❝❧✉✐✲s❡ ❡♥tã♦ q✉❡✱ s❡ F é ✉♠ ❝♦♥❥✉♥t♦ ❢❡❝❤❛❞♦✱ ❡♥tã♦ Cl(F) = F✳

❋♦✐ ✈✐st♦ ❛♥t❡r✐♦r♠❡♥t❡ q✉❡ ✐♥t❡rs❡çã♦ q✉❛❧q✉❡r ❞❡ s✉❜❝♦♥❥✉♥t♦s ❢❡❝❤❛❞♦s ❞❡ ✉♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦✱ é ✉♠ s✉❜❝♦♥❥✉♥t♦ ❢❡❝❤❛❞♦ ❞❡st❡ ❡s♣❛ç♦✳ ❙❡❣✉❡ ❡♥tã♦✱ ❞❛ ❞❡✜♥✐çã♦ ❞❡ ❢❡❝❤♦✱ ❛ s❡❣✉✐♥t❡ ♣r♦♣♦s✐çã♦✿

Pr♦♣♦s✐çã♦ ✶✳✶✳✶✺✳ ❙❡❥❛♠ X ✉♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ ❡ A⊆X✳ ❊♥tã♦✿

(i) A é ✉♠ ❝♦♥❥✉♥t♦ ❢❡❝❤❛❞♦❀

(ii) A⊆A❀ ❡

(iii) A é ♦ ♠❡♥♦r ❢❡❝❤❛❞♦ ❝♦♥t❡♥❞♦ A✳

❉❡✜♥✐çã♦ ✶✳✶✳✶✻✳ ❙❡❥❛♠(X, τ) ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ ❡ A s✉❜❝♦♥❥✉♥t♦ ❞❡ X✳

(i) ❙❡❥❛x∈X✳ ❉❡✜♥❡✲s❡ ♣♦rνx :={U ∈τ :x∈U}♦ ❝♦♥❥✉♥t♦ ❞❛s ✈✐③✐♥❤❛♥ç❛s ❛❜❡rt❛s ❞♦ ♣♦♥t♦ x✳

(ii) x∈X é ♣♦♥t♦ ❛❞❡r❡♥t❡ ❛ A s❡ t♦❞❛ ✈✐③✐♥❤❛♥ç❛ ❛❜❡rt❛ U ❞❡ x ✐♥t❡rs❡❝t❛ ❆✳

❆ s❡❣✉✐♥t❡ ❝❛r❛❝t❡r✐③❛çã♦ ❞❡ ❢❡❝❤♦ é ♠✉✐t♦ út✐❧ ❡ r❡❧❛❝✐♦♥❛ ❢❡❝❤♦ ❝♦♠ ♣♦♥t♦s ❛❞❡r❡♥t❡s✿

Pr♦♣♦s✐çã♦ ✶✳✶✳✶✼✳ ❙❡❥❛♠ X ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦✱ A ⊆ X ❡ B ❜❛s❡ ❞❡ X✳ ❙ã♦ ❡q✉✐✈❛✲

❧❡♥t❡s✿

(i) x∈A✳

(ii) ❚♦❞♦ ❛❜❡rt♦ ❜ás✐❝♦ B ∈B✱ t❛❧ q✉❡ x∈B✱ ✐♥t❡rs❡❝t❛ A✳

(iii) ❚♦❞❛ ✈✐③✐♥❤❛♥ç❛ ❛❜❡rt❛ ❞❡ x ✐♥t❡rs❡❝t❛ A✳

❈♦r♦❧ár✐♦ ✶✳✶✳✶✽✳ ❖ ❢❡❝❤♦ ❞❡ ✉♠ ❝♦♥❥✉♥t♦ A é ♦ ❝♦♥❥✉♥t♦ ❞♦s ♣♦♥t♦s ❛❞❡r❡♥t❡s ❞❡ A✳

❊♠ sí♠❜♦❧♦s✱A ={x∈X :x é ❛❞❡r❡♥t❡ ❛ A}

✶✳✶✳✸ ❋✉♥çõ❡s ❝♦♥tí♥✉❛s ❡ ❤♦♠❡♦♠♦r✜s♠♦s

❉❡✜♥✐çã♦ ✶✳✶✳✶✾✳ ❙❡❥❛ f :X −→Y ✉♠❛ ❢✉♥çã♦ ❡♥tr❡ ❡s♣❛ç♦s t♦♣♦❧ó❣✐❝♦s✳

(i) f é ❝♦♥tí♥✉❛ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❛ ✐♠❛❣❡♠ ✐♥✈❡rs❛ ❞❡ ✉♠ ❛❜❡rt♦ ❞❡ Y é ✉♠ ❛❜❡rt♦

❞❡ X✳

(ii) f é ❛❜❡rt❛ s❡ ❛ ✐♠❛❣❡♠ ❞❡ t♦❞♦ s✉❜❝♦♥❥✉♥t♦ ❛❜❡rt♦ ❞❡ X é ❛❜❡rt♦ ❡♠ Y✳

(iii) f é ❢❡❝❤❛❞❛ s❡ ❛ ✐♠❛❣❡♠ ❞❡ t♦❞♦ s✉❜❝♦♥❥✉♥t♦ ❢❡❝❤❛❞♦ ❞❡ X é ❢❡❝❤❛❞♦ ❡♠ Y✳

(iv) ❯♠❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛ é ✉♠ ❤♦♠❡♦♠♦r✜s♠♦ s❡ ❡❧❛ é ❜✐❥❡t✐✈❛ ❡ s✉❛ ✐♥✈❡rs❛ é ❝♦♥✲

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❖❜s❡r✈❛çã♦ ✶✳✶✳✷✵✳ ◆ã♦ é ❞✐❢í❝✐❧ ✈❡r✐✜❝❛r q✉❡ ❛ ❝♦♠♣♦s✐çã♦ ❞❡ ❢✉♥çõ❡s ❝♦♥tí♥✉❛s é ✉♠❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛✳ ❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱ ❛ ❝♦♠♣♦s✐çã♦ ❞❡ ❤♦♠❡♦♠♦r✜s♠♦s é ✉♠ ❤♦♠❡♦♠♦r✲ ✜s♠♦✳

▲❡♠❛ ✶✳✶✳✷✶✳ ❙❡❥❛ f :X −→Y ✉♠❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛ ❡♥tr❡ ❡s♣❛ç♦s t♦♣♦❧ó❣✐❝♦s✳ ❙❡ f é

❜✐❥❡t✐✈❛✱ ❛s s❡❣✉✐♥t❡s ❛✜r♠❛çõ❡s sã♦ ❡q✉✐✈❛❧❡♥t❡s✿

(1) f é ✉♠ ❤♦♠❡♦♠♦r✜s♠♦✳ (2) f é ❢❡❝❤❛❞❛✳ (3) f é ❛❜❡rt❛✳

❆ s❡❣✉✐♥t❡ ❝❛r❛❝t❡r✐③❛çã♦ ♥♦s ❞✐③ q✉❡✱ ❛ ✜♠ ❞❡ ❝♦♥❝❧✉✐r q✉❡ ❛ ❢✉♥çã♦ é ❝♦♥tí♥✉❛✱ é ♥❡❝❡ssár✐♦ t❡st❛r ❛ ❝♦♥t✐♥✉✐❞❛❞❡ ❞❛ ♠❡s♠❛ ❛♣❡♥❛s ♣❛r❛ ♦s ❛❜❡rt♦s ❜ás✐❝♦s✱ s❡♥❞♦ ♠✉✐t♦ út✐❧ ❡♠ ✈ár✐❛s ❞❡♠♦♥str❛çõ❡s✳

Pr♦♣♦s✐çã♦ ✶✳✶✳✷✷✳ P❛r❛ ✉♠❛ ❢✉♥çã♦f ❞❡ ✉♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦X♥✉♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ Y✱ ❛s s❡❣✉✐♥t❡s ❝♦♥❞✐çõ❡s sã♦ ❡q✉✐✈❛❧❡♥t❡s✿

(i) ❛ ❢✉♥çã♦ f é ❝♦♥tí♥✉❛❀

(ii) ❛ ✐♠❛❣❡♠ ✐♥✈❡rs❛ ❞❡ ❝❛❞❛ ❛❜❡rt♦ ❡♠ ✉♠❛ ❜❛s❡ B ❞❡ Y é ✉♠ ❛❜❡rt♦ ❞❡ X❀

(iii) ❛ ✐♠❛❣❡♠ ✐♥✈❡rs❛ ❞❡ ❝❛❞❛ ❢❡❝❤❛❞♦ ❞❡ Y é ✉♠ ❢❡❝❤❛❞♦ ❞❡ X✳

✶✳✶✳✹ ❆①✐♦♠❛s ❞❡ s❡♣❛r❛çã♦

❉❡✜♥✐çã♦ ✶✳✶✳✷✸✳ ❯♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ X é

(i) T0✱ s❡ ♣❛r❛ ❝❛❞❛ ❞♦✐s ♣♦♥t♦s x, y ∈ X ❞✐st✐♥t♦s✱ ❡①✐st❡ ✉♠ ❛❜❡rt♦ q✉❡ ❝♦♥té♠ ❛♣❡♥❛s

✉♠ ❞❡ss❡s ♣♦♥t♦s✳

(ii) T1✱ s❡ ♣❛r❛ ❝❛❞❛ ❞♦✐s ♣♦♥t♦s x, y ∈X ❞✐st✐♥t♦s✱ ❡①✐st❡ ✉♠ ❛❜❡rt♦ q✉❡ ❝♦♥té♠ x ❡ ♥ã♦

❝♦♥té♠ y ❡ ✉♠ ❛❜❡rt♦ q✉❡ ❝♦♥té♠ y ❡ ♥ã♦ ❝♦♥té♠ x✳

(iii) T2✱ ♦✉ ❍❛✉s❞♦r✛✱ s❡ ♣❛r❛ ❝❛❞❛ ♣❛r ❞❡ ♣♦♥t♦s ❞✐st✐♥t♦sx, y ∈X✱ ❡①✐st❡♠ ❛❜❡rt♦s U

❡ V t❛✐s q✉❡ x∈U✱ y∈V ❡ U ∩V =∅✳

(iv) T3✱ s❡ ♣❛r❛ ❝❛❞❛x∈X ❡ ❝❛❞❛ ❢❡❝❤❛❞♦F ⊆X t❛❧ q✉❡ x6∈F ❡①✐st❡♠ ❛❜❡rt♦s ❞✐s❥✉♥t♦s

U ❝♦♥t❡♥❞♦ x ❡ V ❝♦♥t❡♥❞♦ F✳

(v) T4✱ s❡ ♣❛r❛ ❝❛❞❛ ♣❛r ❞❡ ❢❡❝❤❛❞♦s ❞✐s❥✉♥t♦s✱ F ❡ G✱ ❡①✐st❡♠ ❛❜❡rt♦s ❞✐s❥✉♥t♦s✱ U ❡ V✱

❝♦♥t❡♥❞♦ F ❡ G✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳

❖❜s❡r✈❛çã♦ ✶✳✶✳✷✹✳ ➱ ✐♠❡❞✐❛t♦ ❞❛ ❞❡✜♥✐çã♦ q✉❡ T2 ⇒T1 ⇒T0

❆ ♣r♦♣♦s✐çã♦ s❡❣✉✐♥t❡ ❢♦r♥❡❝❡ ✉♠❛ ❝❛r❛❝t❡r✐③❛çã♦ ❞❡ ❡s♣❛ç♦s T0✳

Pr♦♣♦s✐çã♦ ✶✳✶✳✷✺✳ ❯♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦X é T0 s❡✱ ❡ s♦♠❡♥t❡ s❡✱ {x} 6={y}✱ s❡♠♣r❡

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✶✳✶✳✺ ❈♦♠♣❛❝✐❞❛❞❡

❉❡✜♥✐çã♦ ✶✳✶✳✷✻✳ ❙❡❥❛♠ X ✉♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ ❡ A ⊆X ✉♠ s✉❜❡s♣❛ç♦ ❞❡ X✳ ❙❡❥❛ C ✉♠❛ ❢❛♠í❧✐❛ ❞❡ s✉❜❝♦♥❥✉♥t♦s ❞❡ X✳

(i) ❉✐③❡♠♦s q✉❡ C é ✉♠❛ ❝♦❜❡rt✉r❛ ❞❡ X ✭♦✉ ✉♠ r❡❝♦❜r✐♠❡♥t♦ ❞❡ X✮ s❡ X =S C✳

(ii) ❉✐③❡♠♦s q✉❡ C é ❝♦❜❡rt✉r❛ ❛❜❡rt❛ s❡ s❡✉s ❡❧❡♠❡♥t♦s sã♦ ❛❜❡rt♦s ❞❡ X✳

(iii) ❉✐③❡♠♦s q✉❡ ✉♠❛ s✉❜❢❛♠í❧✐❛ C′ ⊆ C é ✉♠❛ s✉❜❝♦❜❡rt✉r❛ ❞❡ C s❡ S

C′ =X

(iv) ❉✐③❡♠♦s q✉❡ C é ✉♠❛ ❝♦❜❡rt✉r❛ ✜♥✐t❛ s❡ C é ✉♠❛ ❝♦❜❡rt✉r❛ ❝♦♥t❡♥❞♦ ❛♣❡♥❛s ✉♠ ♥ú♠❡r♦ ✜♥✐t♦ ❞❡ ❡❧❡♠❡♥t♦s✳

(v) ❉✐③❡♠♦s q✉❡ ✉♠❛ ❝♦❧❡çã♦ ❞❡ ❛❜❡rt♦s C ❝♦❜r❡ A s❡ A⊆S C✳

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

❛❜❡rt❛ ❞❡ X ♣♦ss✉✐ ✉♠❛ s✉❜❝♦❜❡rt✉r❛ ✜♥✐t❛✳

❚❡♦r❡♠❛ ✶✳✶✳✷✽✳ ❯♠ s✉❜❡s♣❛ç♦ A ⊆ X é ❝♦♠♣❛❝t♦ s❡ ❡ s♦♠❡♥t❡ s❡✱ t♦❞❛ ❢❛♠í❧✐❛ ❞❡

❛❜❡rt♦s ❞❡ X q✉❡ ❝♦❜r❡ A ♣♦ss✉✐ ✉♠❛ s✉❜❢❛♠í❧✐❛ ✜♥✐t❛ q✉❡ ❝♦❜r❡ A✳

❖❜s❡r✈❡ q✉❡✱ s❡ X é ✉♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦✱ ❡♥tã♦ ❛ ✉♥✐ã♦ ✜♥✐t❛ ❞❡ s✉❜❝♦♥❥✉♥t♦s

❝♦♠♣❛❝t♦s ❞❡X é ✉♠ s✉❜❝♦♥❥✉♥t♦ ❝♦♠♣❛❝t♦ ❞❡X✳

❚❡♦r❡♠❛ ✶✳✶✳✷✾✳ ❚♦❞♦ s✉❜❡s♣❛ç♦ ❢❡❝❤❛❞♦ ❞❡ ✉♠ ❡s♣❛ç♦ ❝♦♠♣❛❝t♦ é ❝♦♠♣❛❝t♦✳

▲❡♠❛ ✶✳✶✳✸✵✳ ❙❡❥❛ X ✉♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ ❡ B ✉♠❛ ❜❛s❡ ❞❡ X✳ ❊♥tã♦ X é ❝♦♠♣❛❝t♦

s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❝❛❞❛ ❝♦❜❡rt✉r❛ ❛❜❡rt❛ C ⊆B ♣♦ss✉✐ ✉♠ s✉❜✲r❡❝♦❜r✐♠❡♥t♦ ✜♥✐t♦✳

▲❡♠❛ ✶✳✶✳✸✶✳ ✭❆❧❡①❛♥❞❡r✮ ❙❡❥❛ X ✉♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ ❡ B ✉♠❛ s✉❜✲❜❛s❡ ❞❡ X✳ X é

❝♦♠♣❛❝t♦ s❡✱ ❡ s♦♠❡♥t❡✱ t♦❞♦ r❡❝♦❜r✐♠❡♥t♦ ❛❜❡rt♦ ❞❡X✱ ❝✉❥♦s ❡❧❡♠❡♥t♦s ♣❡rt❡♥❝❡♠ ❛ B✱

❛❞♠✐t❡ ✉♠ s✉❜✲r❡❝♦❜r✐♠❡♥t♦ ✜♥✐t♦✳

❉❡✜♥✐çã♦ ✶✳✶✳✸✷✳ ❉✐③❡♠♦s q✉❡ ✉♠❛ ❢❛♠í❧✐❛ F ={Fs : s ∈ S} t❡♠ ❛ ♣r♦♣r✐❡❞❛❞❡ ❞❛ ✐♥t❡rs❡çã♦ ✜♥✐t❛ ✭♣✐❢✮ s❡✱ ♣❛r❛ ❝❛❞❛S′ S ✜♥✐t♦✱ \

s∈S′

Fs é ✉♠ ❝♦♥❥✉♥t♦ ♥ã♦ ✈❛③✐♦✳

❖ t❡♦r❡♠❛ ❛ s❡❣✉✐r é ♠✉✐t♦ út✐❧ ❡ ❢♦r♥❡❝❡ ✉♠❛ ❝❛r❛❝t❡r✐③❛çã♦ ❞❡ ❡s♣❛ç♦s ❝♦♠✲ ♣❛❝t♦s✳

❚❡♦r❡♠❛ ✶✳✶✳✸✸✳ ❯♠ ❡s♣❛ç♦ X é ❝♦♠♣❛❝t♦ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ t♦❞❛ ❢❛♠í❧✐❛ ❞❡ ❝♦♥❥✉♥t♦s

❢❡❝❤❛❞♦s ❝♦♠ ❛ ♣✐❢ t❡♠ ✐♥t❡rs❡çã♦ ♥ã♦ ✈❛③✐❛✳

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✶✳✷ ❖r❞❡♥s

❖ ❝♦♥❝❡✐t♦ ❞❡ ♦r❞❡♠ ❡stá ♣r❡s❡♥t❡ ♥ã♦ só ♥❛ ▼❛t❡♠át✐❝❛✱ ❝♦♠♦ t❛♠❜é♠ ❡♠ ♥♦ss♦ ❝♦t✐❞✐❛♥♦✱ s❡♠♣r❡ q✉❡ ❢❛③❡♠♦s ❝♦♠♣❛r❛çõ❡s✳ ◆❛ ▼❛t❡♠át✐❝❛✱ ❢r❡q✉❡♥t❡♠❡♥t❡ ❝♦♠♣❛r❛✲ ♠♦s ♦❜❥❡t♦s✱ ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦ ❡❧❡♠❡♥t♦s ❞❡ ✉♠ ❞❛❞♦ ❝♦♥❥✉♥t♦ ✭✏① é ♠❡♥♦r ❞♦ q✉❡ ②✑✮✱ q✉❡ ♣♦❞❡♠ s❡r ❝♦♥❥✉♥t♦s✱ ✐♥❝❧✉s✐✈❡ ✭✏❆ ❡stá ❝♦♥t✐❞♦ ❡♠ ❇✑✮✳ P♦r s❡r ✉♠ ❛ss✉♥t♦ ♠✉✐t♦ ❛♠♣❧♦ ❡ ❡♥✈♦❧✈❡r ❞✐✈❡rs♦s ♦❜❥❡t♦s ♠❛t❡♠át✐❝♦s✱ ❛ ❚❡♦r✐❛ ❞❛s ❖r❞❡♥s ❢♦✐ ❢♦r♠❛❧✐③❛❞❛ ❡ ❞❡s❡♥✈♦❧✈✐❞❛ ❡ ♣♦ss✉✐ ❛♣❧✐❝❛çõ❡s ❡♠ ❞✐✈❡rs❛s ár❡❛s✱ ❡s♣❡❝✐❛❧♠❡♥t❡ ♥❛ ❚❡♦r✐❛ ❞❡ ❘❡t✐❝✉❧❛✲ ❞♦s ❡ ♥❛ ▲ó❣✐❝❛✳ ◆❡st❛ s❡çã♦ ❡stã♦ ♣r❡s❡♥t❡s ❛♣❡♥❛s ❝♦♥❝❡✐t♦s ❜ás✐❝♦s ❞❡st❛ ❚❡♦r✐❛ ❞❛s ❖r❞❡♥s✳ ❯♠ ❡st✉❞♦ ♠❛✐s ❛♣r♦❢✉♥❞❛❞♦ s♦❜r❡ ♦r❞❡♥s ♣♦❞❡ s❡r ✈✐st♦ ❡♠ ❬❉P✾✷❪✳

✶✳✷✳✶ ❈♦♥❝❡✐t♦s ❜ás✐❝♦s s♦❜r❡ ♦r❞❡♥s

❉❡✜♥✐çã♦ ✶✳✷✳✶✳ ❙❡❥❛L ✉♠ ❝♦♥❥✉♥t♦✳ ❯♠❛ r❡❧❛çã♦ ❜✐♥ár✐❛✱≤✱ ❡♠ ▲ é ✉♠❛ ♣ré✲♦r❞❡♠

❡♠ L s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ♣❛r❛ t♦❞♦s a, b, c∈L t❡♠✲s❡

(i) a≤a ✭r❡✢❡①✐✈✐❞❛❞❡✮❀

(ii) a≤b ❡ b ≤c ⇒ a≤c ✭tr❛♥s✐t✐✈✐❞❛❞❡✮✳

❯♠❛ ♣ré✲♦r❞❡♠ ≤ ❡♠ L q✉❡ s❛t✐s❢❛③✱ ♣❛r❛ t♦❞♦s a, b∈L✿

(iii) a≤b ❡ b ≤a ⇒ a=b ✭❛♥t✐s✐♠❡tr✐❛✮✱

é ❝❤❛♠❛❞❛ ♦r❞❡♠ ♣❛r❝✐❛❧ ❡♠ ▲✳ ❉✐③✲s❡ ❡♥tã♦ q✉❡hL,≤i é ✉♠ ❝♦♥❥✉♥t♦ ♣❛r❝✐❛❧♠❡♥t❡ ♦r❞❡♥❛❞♦✳

❯♠❛ ♦r❞❡♠ ♣❛r❝✐❛❧ ≤ ❡♠ L é ❞✐t❛ ❧✐♥❡❛r s❡ t♦❞♦s ♦s ❡❧❡♠❡♥t♦s ❞❡ L ♣♦❞❡♠ s❡r

❝♦♠♣❛r❛❞♦s ♣♦r ❡st❛ ♦r❞❡♠✱ ♦✉ s❡❥❛✱ s❡ ♣❛r❛a, b∈L✈❛❧❡ a≤b ♦✉ b≤a✳

❊①❡♠♣❧♦ ✶✳✷✳✷✳

(a) ❆ ❞✐✈✐sã♦✱| ✱ ❡♠ Z∗ é ✉♠❛ ♣ré✲♦r❞❡♠✱ ♣♦✐s✿

(i) a|a❀

(ii) a|b e b|c⇒a|c✳

➱ ❝❧❛r♦ q✉❡ ❡st❛ ♦r❞❡♠ ♥ã♦ é ❛♥t✐s✐♠étr✐❝❛✳

(b) hN, ≤i é ✉♠ ❝♦♥❥✉♥t♦ ♣❛r❝✐❛❧♠❡♥t❡ ♦r❞❡♥❛❞♦✱ ♣♦✐s ♣❛r❛ ❛ r❡❧❛çã♦ ≤✉s✉❛❧ ❡♠N ✈❛❧❡♠

r❡✢❡①✐✈✐❞❛❞❡✱ tr❛♥s✐t✐✈✐❞❛❞❡ ❡ ❛♥t✐s✐♠❡tr✐❛✳

(c) ❙❡❥❛ (X, τ) ✉♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦✳ ➱ ❢á❝✐❧ ✈❡r q✉❡ hτ,⊆i é ✉♠ ❝♦♥❥✉♥t♦ ♣❛r❝✐❛❧♠❡♥t❡ ♦r❞❡♥❛❞♦✳ P♦st❡r✐♦r♠❡♥t❡✱ s❡rá ✈✐st♦ q✉❡ τ é ✉♠❛ á❧❣❡❜r❛ ❞❡ ❍❡②t✐♥❣✳

(d) ❙❡❥❛ X ✉♠ ❝♦♥❥✉♥t♦✳ ❊♥tã♦✱ ❛ ♦r❞❡♠ ❞❛❞❛ ♣❡❧❛ ✐♥❝❧✉sã♦ ❡♠ P(X) é ✉♠❛ ♦r❞❡♠

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P❛r❛ hL, ≤ i ❝♦♥❥✉♥t♦ ♣❛r❝✐❛❧♠❡♥t❡ ♦r❞❡♥❛❞♦✱ 1 ✭♦✉ ⊤✮ ❡ 0 ✭♦✉ ⊥✮ s❡♠♣r❡

❞❡♥♦t❛rã♦max L ❡ min L ✭q✉❛♥❞♦ ❡①✐st✐r❡♠✮✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳

❖❜s❡r✈❛çã♦ ✶✳✷✳✸✳ ❙❡❥❛ hL, ≤ i ✉♠ ❝♦♥❥✉♥t♦ ♣❛r❝✐❛❧♠❡♥t❡ ♦r❞❡♥❛❞♦✳ ➱ ❢á❝✐❧ ✈❡r q✉❡ hL, ≤opi t❛♠❜é♠ é ✉♠ ❝♦♥❥✉♥t♦ ♣❛r❝✐❛❧♠❡♥t❡ ♦r❞❡♥❛❞♦ ✱ ♦♥❞❡

∀x, y ∈L, x≤opy ss❡ yx

❙❡❥❛ ❛❣♦r❛ ψ ✉♠❛ ❛✜r♠❛çã♦ ❢♦r♠✉❧❛❞❛ ♥❛ ❧✐♥❣✉❛❣❡♠ ❞❛s ♦r❞❡♥s✳ ❆ ❛✜r♠❛çã♦ ❞✉❛❧ ❞❡ ψ✱ ψop✱ é ❞❛❞❛ ♣♦rψ✱ ❛♣❡♥❛s s✉❜st✐t✉✐♥❞♦ ❝❛❞❛ ♦❝♦rrê♥❝✐❛ ❞❡ ♣♦r

❖ ♣r✐♥❝í♣✐♦ ❞❛ ❞✉❛❧✐❞❛❞❡ ♣r♦❝❧❛♠❛ ♦ s❡❣✉✐♥t❡✿ ❙❡❥❛ ψ ✉♠❛ ❛✜r♠❛çã♦ ❢♦r♠✉❧❛❞❛ ♥❛ ❧✐♥✲

❣✉❛❣❡♠ ❞❛s ♦r❞❡♥s ♣❛r❝✐❛✐s ❡ ✈á❧✐❞❛ ❡♠ q✉❛❧q✉❡r ❝♦♥❥✉♥t♦ ♣❛r❝✐❛❧♠❡♥t❡ ♦r❞❡♥❛❞♦✱ ❡♥tã♦ ❛ ❛✜r♠❛çã♦ ❞✉❛❧✱ ψop✱ t❛♠❜é♠ é ✈á❧✐❞❛ ❡♠ q✉❛❧q✉❡r ❝♦♥❥✉♥t♦ ♣❛r❝✐❛❧♠❡♥t❡ ♦r❞❡♥❛❞♦✳ ❊♠ s❡❣✉✐❞❛ s❡rá ❢❡✐t♦ ✉s♦ ❝♦♥st❛♥t❡ ❞❡st❡ s✐♠♣❧❡s ♣r✐♥❝í♣✐♦✱ s❡♠ ♠✉✐t❛s ❡①♣❧✐❝❛çõ❡s ❛❞✐✲ ❝✐♦♥❛✐s✳ ❱❛❧❡ r❡ss❛❧t❛r q✉❡ s✉❛ ❞❡♠♦♥str❛çã♦ ♣♦❞❡ s❡r ❢❡✐t❛ ♣♦r ■♥❞✉çã♦ ♥❛ ❈♦♠♣❧❡①✐❞❛❞❡ ❞❛s ❋ór♠✉❧❛s✳

✶✳✷✳✷ ❈♦♥❥✉♥t♦s ❝r❡s❝❡♥t❡s ❡ ❞❡❝r❡s❝❡♥t❡s

P❛r❛ hX,≤i✉♠ ❝♦♥❥✉♥t♦ ♣ré✲♦r❞❡♥❛❞♦ ✭✐✳❡✳✱ ≤ ✉♠❛ ♣ré✲♦r❞❡♠✮ ❡U ✉♠ s✉❜❝♦♥✲

❥✉♥t♦ ❞❡ X✱ ❞❡♥♦t❛r✲s❡✲á✿

 

↑U :={x∈X|∃y∈U(y≤x)}

↓U :={x∈X|∃y∈U(x≤y)}

❉❡✜♥✐çã♦ ✶✳✷✳✹✳ ❙❡❥❛ hX,≤i ✉♠ ❝♦♥❥✉♥t♦ ♣❛r❝✐❛❧♠❡♥t❡ ♦r❞❡♥❛❞♦ ❡ U ✉♠ s✉❜❝♦♥❥✉♥t♦

❞❡ X✳

(i) ❉✐③✲s❡ q✉❡ U é ❝r❡s❝❡♥t❡ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ U =↑U✳

(ii) ❉✐③✲s❡ q✉❡ U é ❞❡❝r❡s❝❡♥t❡ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ U =↓U✳

P❛r❛{b} ⊆X✱ s❡rá ✉t✐❧✐③❛❞❛ ❛ ♥♦t❛çã♦↑b✭↓b✮ ❛♦ ✐♥✈és ❞❡↑ {b}✭r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ↓ {b}✮✳

❖❜s❡r✈❡ q✉❡U ⊆↑U ❡U ⊆↓U ✈❛❧❡♠ s❡♠♣r❡✳ ❉❡ss❛ ❢♦r♠❛✱ ♣❛r❛ ♠♦str❛r q✉❡ ✉♠

❞❛❞♦ ❝♦♥❥✉♥t♦ é ❝r❡s❝❡♥t❡ ♦✉ ❞❡❝r❡s❝❡♥t❡✱ é ♣r❡❝✐s♦ ✈❡r✐✜❝❛r ❛♣❡♥❛s ✉♠❛ ✐♥❝❧✉sã♦✳

❈♦♥s✐❞❡r❛♥❞♦ ❛ r❡t❛ r❡❛❧ ❝♦♠ ❛ ♦r❞❡♠ ≤ ✉s✉❛❧✱ ❝♦♠♦ ❡①❡♠♣❧♦s ❞❡ ❝♦♥❥✉♥t♦s ❝r❡s❝❡♥t❡ ❡ ❞❡❝r❡s❝❡♥t❡ t❡♠✲s❡ ♦s s✉❜❝♦♥❥✉♥t♦s R+✱ ❞♦s ♥ú♠❡r♦s r❡❛✐s ♥ã♦✲♥❡❣❛t✐✈♦s✱ ❡ R−✱ ❞♦s ♥ú♠❡r♦s r❡❛✐s ♥ã♦✲♣♦s✐t✐✈♦s✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳

❋❛t♦ ✶✳✷✳✺✳ ❙❡❥❛♠(X,≤) ❝♦♥❥✉♥t♦ ♣ré✲♦r❞❡♥❛❞♦ ❡ A=↑A✳ ❊♥tã♦

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✶✵

❊♠ ♣❛rt✐❝✉❧❛r✱ s❡(X,≤)❢♦r ♦r❞❡♠ ♣❛r❝✐❛❧✱ t❡♠✲s❡ q✉❡ ♦ ❝♦♠♣❧❡♠❡♥t❛r ❞❡ ✉♠ s✉❜❝♦♥❥✉♥t♦

❝r❡s❝❡♥t❡ ❞❡ X é ✉♠ s✉❜❝♦♥❥✉♥t♦ ❞❡❝r❡s❝❡♥t❡ ❞❡ X ❡ ✈✐❝❡✲✈❡rs❛✳

Pr♦✈❛✿

❙❡❥❛ U ⊆ X ❝r❡s❝❡♥t❡✱ ✐✳❡✳✱ U =↑U✳ ❙❡rá ♠♦str❛❞♦ q✉❡ X\ ↑U =↓(X\ ↑ U)✳

❈♦♠♦ s❡♠♣r❡ ✈❛❧❡ A⊆↓A✱ ♣❛r❛ t♦❞♦ A⊆X✱ t❡♠✲s❡ q✉❡ X\ ↑U ⊆↓(X\ ↑U)✳ ❱❡❥❛♠♦s

❛ ♦✉tr❛ ✐♥❝❧✉sã♦✳ ❙❡❥❛x ∈↓(X\ ↑ U)✳ ▲♦❣♦✱ ❡①✐st❡ z ∈ X\ ↑U t❛❧ q✉❡ x ≤z✳ P♦rt❛♥t♦✱ x 6∈↑ U✱ ❝❛s♦ ❝♦♥trár✐♦✱ ♣♦r ❞❡✜♥✐çã♦✱ z ∈↑ U✳ ❆ss✐♠✱ x ∈ X\ ↑ U✳ ❉❡ss❛ ❢♦r♠❛✱

↓(X\ ↑U)⊆X\ ↑U✳

Pr♦♣♦s✐çã♦ ✶✳✷✳✻✳ ❙❡❥❛ (X,≤) ❝♦♥❥✉♥t♦ ♣❛r❝✐❛❧♠❡♥t❡ ♦r❞❡♥❛❞♦✳ ❊♥tã♦✱

(i) ✉♥✐ã♦ ❞❡ s✉❜❝♦♥❥✉♥t♦s ❝r❡s❝❡♥t❡s ✭❞❡❝r❡s❝❡♥t❡s✮ ❞❡ X é ✉♠ s✉❜❝♦♥❥✉♥t♦ ❝r❡s❝❡♥t❡ ✭r❡✲

s♣❡❝t✐✈❛♠❡♥t❡✱ ❞❡❝r❡s❝❡♥t❡✮ ❞❡ X❀

(ii) ✐♥t❡rs❡çã♦ ❞❡ s✉❜❝♦♥❥✉♥t♦s ❝r❡s❝❡♥t❡s ✭❞❡❝r❡s❝❡♥t❡s✮ ❞❡ X é ✉♠ s✉❜❝♦♥❥✉♥t♦ ❝r❡s❝❡♥t❡

✭r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❞❡❝r❡s❝❡♥t❡✮ ❞❡ X✳

Pr♦✈❛✿

(i) ❙❡❥❛ {Ui}i∈I ❢❛♠í❧✐❛ ❞❡ s✉❜❝♦♥❥✉♥t♦s ❝r❡s❝❡♥t❡s ❞❡X✳ x ∈↑[

i∈I

Ui ⇒ ∃y∈ [

i∈I

Ui t❛❧ q✉❡ y ≤x⇒ ∃j ∈ I t❛❧ q✉❡ y∈Uj ❡ y≤x⇒x∈ Uj✱

♣♦✐s Uj é ❝r❡s❝❡♥t❡ ⇒x∈ [

i∈I

Ui ⇒↑ [

i∈I

Ui ⊆ [

i∈I

Ui

P♦rt❛♥t♦ [ i∈I

Ui =↑ [

i∈I

Ui✳

(ii) ❊st❡ ✐t❡♠ é ❞❡♠♦♥str❛❞♦ ❞❡ ❢♦r♠❛ ❛♥á❧♦❣❛ ❛♦ ❛♥t❡r✐♦r✳

P❛r❛ ✉♠ ❝♦♥❥✉♥t♦ ♣❛r❝✐❛❧♠❡♥t❡ ♦r❞❡♥❛❞♦ (X, τ)✱ ♥ã♦ é ❞✐❢í❝✐❧ ♠♦str❛r q✉❡ ❛ ❛♣❧✐✲

❝❛çã♦↑:P(X)→ P(X)✱ A7→↑A✱ s❛t✐s❢❛③ ♦s ❛①✐♦♠❛s ❞❡ ❚❛rs❦✐ ❧✐st❛❞♦s ❛ s❡❣✉✐r✿

(i) A ⊆↑A❀

(ii) A ⊆B ⇒↑A⊆↑B❀

(iii) ↑↑A =↑A✳

❖ ♠❡s♠♦ é ✈á❧✐❞♦ ♣❛r❛ ↓: P(X) → P(X)✳ ❙❡♥❞♦ ❛ss✐♠✱ ↑,↓ sã♦ ♦♣❡r❛❞♦r❡s ❞❡ ❚❛rs❦✐✳ ❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱ ❛s ❝♦❧❡çõ❡s {A ⊆ X : A =↑A} ❡ {A ⊆ X : A =↓A} sã♦ T

−❡str✉t✉r❛s ✭✈❡❥❛ ❬❉P✾✷❪✮✳ ❖ ✐t❡♠ (ii) ❞❛ ♣r♦♣♦s✐çã♦ ❛❝✐♠❛ é ✉♠❛ ❝♦♥s❡q✉ê♥❝✐❛

Figure

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