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❯♥✐✈❡rs✐❞❛❞❡ ❊st❛❞✉❛❧ P❛✉❧✐st❛ ✏❏ú❧✐♦ ❞❡ ▼❡sq✉✐t❛ ❋✐❧❤♦✑ ■♥st✐t✉t♦ ❞❡ ●❡♦❝✐ê♥❝✐❛s ❡ ❈✐ê♥❝✐❛s ❊①❛t❛s

❈â♠♣✉s ❞❡ ❘✐♦ ❈❧❛r♦

❯♠ ❡st✉❞♦ s♦❜r❡ ❛ ❝❧❛ss✐✜❝❛çã♦ t♦♣♦❧ó❣✐❝❛

❞❛s s✉♣❡r❢í❝✐❡s✳

❆♥❛ ❋❧á✈✐❛ ▼❛r✐❛♥♦ ❞❡ ❙♦✉s❛

❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ Pr♦❣r❛♠❛ ❞❡ Pós✲ ●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ▼❡str❡

❖r✐❡♥t❛❞♦r❛

Pr♦❢❛✳ ❉r❛✳ ❚❤❛ís ❋❡r♥❛♥❞❛ ▼❡♥❞❡s ▼♦♥✐s

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Sousa, Ana Flávia Mariano de

Um estudo sobre a classificação topológica das superfícies / Ana Flávia Mariano de Sousa. - Rio Claro, 2016

83 f. : il., figs.

Dissertação (mestrado) - Universidade Estadual Paulista, Instituto de Geociências e Ciências Exatas

Orientador: Thaís Fernanda Mendes Monis

1. Topologia. 2. Classificação das superfícies compactas. 3. Espaço topológico. 4. Soma conexa de superfícies. I. Título.

514 S725e

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❚❊❘▼❖ ❉❊ ❆P❘❖❱❆➬➹❖

❆♥❛ ❋❧á✈✐❛ ▼❛r✐❛♥♦ ❞❡ ❙♦✉s❛

❯♠ ❡st✉❞♦ s♦❜r❡ ❛ ❝❧❛ss✐❢✐❝❛çã♦ t♦♣♦❧ó❣✐❝❛ ❞❛s

s✉♣❡r❢í❝✐❡s✳

❉✐ss❡rt❛çã♦ ❛♣r♦✈❛❞❛ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ▼❡str❡ ♥♦ ❈✉rs♦ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ❞♦ ■♥st✐t✉t♦ ❞❡ ●❡♦❝✐ê♥❝✐❛s ❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❊st❛❞✉❛❧ P❛✉❧✐st❛ ✏❏ú❧✐♦ ❞❡ ▼❡sq✉✐t❛ ❋✐❧❤♦✑✱ ♣❡❧❛ s❡❣✉✐♥t❡ ❜❛♥❝❛ ❡①❛♠✐♥❛❞♦r❛✿

Pr♦❢❛✳ ❉r❛✳ ❚❤❛ís ❋❡r♥❛♥❞❛ ▼❡♥❞❡s ▼♦♥✐s ❖r✐❡♥t❛❞♦r❛

Pr♦❢✳ ❉r✳ ◆❡❧s♦♥ ❆♥tô♥✐♦ ❙✐❧✈❛ ❯❋▲❆ ✲ ▲❛✈r❛s✭▼●✮

Pr♦❢✳ ❉r✳ ❙ér❣✐♦ ❚s✉②♦s❤✐ ❯r❛ ❆✉tô♥♦♠♦ ✲ ❘✐♦ ❈❧❛r♦✭❙P✮

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❆❣r❛❞❡❝✐♠❡♥t♦s

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

❆♦s ♠❡✉s ♣❛✐s✱ ▼❛r✐❛ ❡ ❏♦sé✱ ♠❡✉ ✐♥✜♥✐t♦ ❛❣r❛❞❡❝✐♠❡♥t♦✳ P♦r t♦❞♦ ❛♠♦r ❡ t❡♠♣♦ ❞❛ ✈✐❞❛ ❞❡ ✈♦❝ês ❞❡❞✐❝❛❞♦ ❛ ♠✐♠✳ P♦r ❛❝r❡❞✐t❛r❡♠ ♥❛ ♠✐♥❤❛ ❝❛♣❛❝✐❞❛❞❡✱ ♣♦r t♦❞♦ ❛♣♦✐♦ ♥♦s ♠♦♠❡♥t♦s q✉❡ ♣r❡❝✐s❡✐✳ ❖❜r✐❣❛❞❛ ♣♦r t♦❞♦ ❛♠♦r✳

❆♦s ♠❡✉s q✉❡r✐❞♦s ✐r♠ã♦s ❆❧❡① ❙❛♥❞r♦✱ ▲✉❝✐♥é✐❛ ❡ ❏♦ã♦ P❛✉❧♦✱ ♣♦r s❡r❡♠ ♠❡✉s ❡t❡r♥♦s ❝♦♠♣❛♥❤❡✐r♦s ♥♦ ❝❛♠✐♥❤♦ ❞❛ ✈✐❞❛ ❡ ♣♦r s❡♠♣r❡ ❡st❛r❡♠ ♣r♦♥t♦s ♣❛r❛ ♠❡ ❛❥✉❞❛r✳

❆♦s ♠❡✉s s♦❜r✐♥❤♦s✱ ♣♦r t♦r♥❛r❡♠ ♠❡✉s ❞✐❛s ♠❛✐s ❛❧❡❣r❡s✱ ♠❡s♠♦ ❡♠ ♠❡✐♦ ❛s ❞✐✜❝✉❧❞❛❞❡s✳

❆♦ ♠❡✉ ♥❛♠♦r❛❞♦ ❘✉❜❡♥s✱ ♣♦r ❡st❛r s❡♠♣r❡ ❛♦ ♠❡✉ ❧❛❞♦✱ ♣♦r t♦❞♦ ❛♠♦r✱ ❛♣♦✐♦ ❡ ❝♦♠♣❛♥❤❡r✐s♠♦✳

➚ ♠✐♥❤❛ ♦r✐❡♥t❛❞♦r❛ ❡ ❡①❡♠♣❧♦ ♣r♦✜ss✐♦♥❛❧ ❚❤❛ís ❋❡r♥❛♥❞❛ ▼❡♥❞❡s ▼♦♥✐s✱ ♣♦r t♦❞❛ ❞❡❞✐❝❛çã♦ ❡ ❡♥s✐♥❛♠❡♥t♦s✳

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✧P♦r ✈❡③❡s s❡♥t✐♠♦s q✉❡ ❛q✉✐❧♦ q✉❡ ❢❛③❡♠♦s ♥ã♦ é s❡♥ã♦ ✉♠❛ ❣♦t❛ ❞❡ á❣✉❛ ♥♦ ♠❛r✳ ▼❛s ♦ ♠❛r s❡r✐❛ ♠❡♥♦r s❡ ❧❤❡ ❢❛❧t❛ss❡ ✉♠❛ ❣♦t❛✳✧

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❘❡s✉♠♦

◆❡ss❡ tr❛❜❛❧❤♦✱ ❝♦♥s✐❞❡r❛♠♦s ♦ ❝♦♥❝❡✐t♦ ❞❡ ✈❛r✐❡❞❛❞❡s t♦♣♦❧ó❣✐❝❛s✳ P♦ré♠✱ ♥♦s ❡s♣❡❝✐❛❧✐③❛♠♦s ♥❛s ✈❛r✐❡❞❛❞❡s ❝♦♥❡①❛s ❞❡ ❞✐♠❡♥sã♦ 2✱ ❛s ❝❤❛♠❛❞❛s s✉♣❡r❢í❝✐❡s✳

◆♦ss♦ ♦❜❥❡t✐✈♦ é ♦ ❡st✉❞♦ ❞❛ ❝❧❛ss✐✜❝❛çã♦ t♦♣♦❧ó❣✐❝❛ ❞❛s s✉♣❡r❢í❝✐❡s ❝♦♠♣❛❝t❛s✳ P❛r❛ ✐st♦✱ ❡♥✉♥❝✐❛♠♦s ❡ ❞❡♠♦♥str❛♠♦s ♦ ❚❡♦r❡♠❛ ❞❡ ❝❧❛ss✐✜❝❛çã♦ ❞❛s s✉♣❡r❢í❝✐❡s ❝♦♠✲ ♣❛❝t❛s✳ ❉❡st❛ ♠❛♥❡✐r❛✱ ♠♦str❛♠♦s q✉❡ t♦❞❛ s✉♣❡r❢í❝✐❡ ❝♦♠♣❛❝t❛ ♦r✐❡♥tá✈❡❧ é ❤♦♠❡♦✲ ♠♦r❢❛ à ❡s❢❡r❛ ♦✉ ❛ ✉♠❛ s♦♠❛ ❝♦♥❡①❛ ❞❡ t♦r♦s✱ ❡ q✉❡ t♦❞❛ s✉♣❡r❢í❝✐❡ ❝♦♠♣❛❝t❛ ♥ã♦✲ ♦r✐❡♥tá✈❡❧ é ❤♦♠❡♠♦r❢❛ ❛ ✉♠❛ s♦♠❛ ❝♦♥❡①❛ ❞❡ ♣❧❛♥♦s ♣r♦❥❡t✐✈♦s✳

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❆❜str❛❝t

■♥ t❤✐s ✇♦r❦✱ ✇❡ ❝♦♥s✐❞❡r t❤❡ ❝♦♥❝❡♣t ♦❢ ❛ t♦♣♦❧♦❣✐❝❛❧ ♠❛♥✐❢♦❧❞✳ ❍♦✇❡✈❡r✱ ✇❡ ❢♦❝✉s ♦♥ t❤❡ ❝♦♥♥❡❝t❡❞ 2✲❞✐♠❡♥s✐♦♥❛❧ ♠❛♥✐❢♦❧❞s✱ t❤❡ s♦✲❝❛❧❧❡❞ s✉r❢❛❝❡s✳

❖✉r ❣♦❛❧ ✐s t❤❡ st✉❞② ♦❢ t❤❡ t♦♣♦❧♦❣✐❝❛❧ ❝❧❛ss✐✜❝❛t✐♦♥ ♦❢ t❤❡ ❝♦♠♣❛❝t s✉r❢❛❝❡s✳ ■♥ t❤✐s ❞✐r❡❝t✐♦♥✱ ✇❡ st❛t❡ ❛♥❞ ♣r♦✈❡ t❤❡ ❝❧❛ss✐✜❝❛t✐♦♥ ❚❤❡♦r❡♠ ♦❢ ❝♦♠♣❛❝t s✉r❢❛❝❡s✳ ❚❤❛t ✐s✱ ✇❡ s❤♦✇ t❤❛t ❡✈❡r② ♦r✐❡♥t❛❜❧❡ ❝♦♠♣❛❝t s✉r❢❛❝❡ ✐s ❤♦♠❡♦♠♦r♣❤✐❝ t♦ t❤❡ s♣❤❡r❡ ♦r t♦ ❛ ❝♦♥♥❡❝t❡❞ s✉♠ ♦❢ t♦r✉s✱ ❛♥❞ ❡✈❡r② ♥♦♥✲♦r✐❡♥t❛❜❧❡ ❝♦♠♣❛❝t s✉r❢❛❝❡ ✐s ❤♦♠❡♦♠♦r♣❤✐❝ t♦ ❛ ❝♦♥♥❡❝t❡❞ s✉♠ ♦❢ ♣r♦❥❡❝t✐✈❡ ♣❧❛♥❡s✳

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▲✐st❛ ❞❡ ❋✐❣✉r❛s

✹✳✶ ❋❛✐①❛ ❞❡ ▼ö❜✐✉s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✺✳✶ ❚♦r♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✺✳✷ P❧❛♥♦ ♣r♦❥❡t✐✈♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✺✳✸ P❧❛♥♦ Pr♦❥❡t✐✈♦ ❝♦♥té♠ ✉♠❛ ❢❛✐①❛ ❞❡ ▼ö❜✐✉s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✻✳✶ ❙♦♠❛ ❈♦♥❡①❛ ❞❡ ❞♦✐s t♦r♦s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✻✳✷ ❙♦♠❛ ❈♦♥❡①❛ ❞❡ três t♦r♦s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✻✳✸ ❈♦♥str✉çã♦ ❞♦ ♣❧❛♥♦ ♣r♦❥❡t✐✈♦ ❝♦♠♦ ❡s♣❛ç♦ q✉♦❝✐❡♥t❡ ❞❡ ✉♠ ♣♦❧í❣♦♥♦

❞❡ ❞♦✐s ❧❛❞♦s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✻✳✹ ❙♦♠❛ ❈♦♥❡①❛ ❞❡ ❞♦✐s ♣❧❛♥♦s ♣r♦❥❡t✐✈♦s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✻✳✺ ❈♦♥str✉çã♦ ❞❛ s♦♠❛ ❝♦♥❡①❛ ❞❡ três ♣❧❛♥♦s ♣r♦❥❡t✐✈♦s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✻✳✻ ❙♦♠❛ ❈♦♥❡①❛ ❞❡ ▼ ❡ ❛ ❡s❢❡r❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✻✳✼ ❈♦♥str✉çã♦ ❞❛ ❡s❢❡r❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✻✳✽ ●❛rr❛❢❛ ❞❡ ❑❧❡✐♥✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✻✳✾ ❖ ❝♦♠♣❧❡♠❡♥t♦ ❞♦ ✐♥t❡r✐♦r ❞❡ Di ❡♠ Si é ❤♦♠❡♦♠♦r❢♦ ❛ ✉♠❛ ❢❛✐①❛ ❞❡

▼ö❜✐✉s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✻✳✶✵ ❈♦rt❡ ❞❛ ●❛rr❛❢❛ ❞❡ ❑❧❡✐♥ ♣❛r❛ ♦❜t❡r ❞✉❛s ❢❛✐①❛s ❞❡ ▼ö❜✐✉s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✻✳✶✶ ❆ s♦♠❛ ❝♦♥❡①❛ ❞❡ ❞♦✐s ♣❧❛♥♦s ♣r♦❥❡t✐✈♦s é ❛ ❣❛rr❛❢❛ ❞❡ ❑❧❡✐♥✳ ✳ ✳ ✳ ✳ ✳ ✹✹ ✽✳✶ ❚✐♣♦s ✐♥❛❞✐♠✐ssí✈❡✐s ❞❡ ✐♥t❡rs❡❝çã♦ ❞❡ tr✐â♥❣✉❧♦s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼ ✽✳✷ ❚r✐❛♥❣✉❧❛r✐③❛çã♦ ❞♦ ♣❧❛♥♦ ♣r♦❥❡t✐✈♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽ ✽✳✸ ❚r✐❛♥❣✉❧❛r✐③❛çã♦ ❞♦ t♦r♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾ ✾✳✶ ❊❧✐♠✐♥❛çã♦ ❞❡ ❆r❡st❛s ❆❞❥❛❝❡♥t❡s ❞❡ Pr✐♠❡✐r❛ ❊s♣é❝✐❡✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹ ✾✳✷ ❚r❛♥s❢♦r♠❛çã♦ ❡♠ ✉♠ ♣♦❧í❣♦♥♦ ❝♦♠ t♦❞♦s ♦s ✈ért✐❝❡s ♥✉♠❛ ♠❡s♠❛

❝❧❛ss❡ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺ ✾✳✸ ❈♦♠♦ ❢❛③❡r ❛❞❥❛❝❡♥t❡s t♦❞♦ ♣❛r ❞❡ s❡❣✉♥❞❛ ❡s♣é❝✐❡✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✻ ✾✳✹ P♦❧í❣♦♥♦ ❡♠ q✉❡ ❛s ❛r❡st❛s ❞❡ ♣r✐♠❡✐r❛ ❡s♣é❝✐❡ ♥ã♦ ❡stã♦ s❡♣❛r❛❞❛s ♣♦r

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(18)

❙✉♠ár✐♦

✶ ■♥tr♦❞✉çã♦ ✶✾

✷ Pr❡❧✐♠✐♥❛r❡s ✷✶

✸ ❱❛r✐❡❞❛❞❡s t♦♣♦❧ó❣✐❝❛s ✷✺

✹ ❱❛r✐❡❞❛❞❡s ❜✐❞✐♠❡♥s✐♦♥❛✐s ✲ ♦r✐❡♥tá✈❡✐s ❡ ♥ã♦✲♦r✐❡♥tá✈❡✐s✳ ✷✾

✺ ❊①❡♠♣❧♦s ❞❡ s✉♣❡r❢í❝✐❡s ❝♦♠♣❛❝t❛s✳ ✸✶

✻ ❙♦♠❛ ❝♦♥❡①❛ ✸✼

✼ P❛r❛ ❝❛❞❛ s✉♣❡r❢í❝✐❡ ❝♦♠♣❛❝t❛✱ ✉♠❛ ✏♣❛❧❛✈r❛✑✳ ✹✺

✽ ❚r✐❛♥❣✉❧❛r✐③❛çã♦ ❞❡ ❙✉♣❡r❢í❝✐❡s ❈♦♠♣❛❝t❛s✳ ✹✼

✾ ❚❡♦r❡♠❛ ❞❛ ❈❧❛ss✐✜❝❛çã♦ ❞❡ ❙✉♣❡r❢í❝✐❡s ❈♦♠♣❛❝t❛s✳ ✺✶

✶✵ ❆ ❈❛r❛❝t❡ríst✐❝❛ ❞❡ ❊✉❧❡r ❞❡ ✉♠❛ ❙✉♣❡r❢í❝✐❡✳ ✼✸

(19)
(20)

✶ ■♥tr♦❞✉çã♦

❖ ❝♦♥❝❡✐t♦ ❢♦r♠❛❧ ❞❡ s✉♣❡r❢í❝✐❡ ✭♦✉ ✈❛r✐❡❞❛❞❡ ❝♦♥❡①❛ ❜✐❞✐♠❡♥s✐♦♥❛❧✮ ♣♦❞❡ s❡r ✈✐st♦ ❝♦♠♦ ✉♠❛ ❛❜str❛çã♦ ♣❛r❛ ❛ ♥♦ss❛ ❡①♣❡r✐ê♥❝✐❛ ❢❛♠✐❧✐❛r ❝♦♠ s✉♣❡r❢í❝✐❡s ❢❡✐t❛s ❞❡ ♣❛♣❡❧✱ ❢♦❧❤❛ ♠❡tá❧✐❝❛✱ ♣❧ást✐❝♦✱ ♦✉ q✉❛❧q✉❡r ♦✉tr♦ ♠❛t❡r✐❛❧ ✜♥♦ ❡ ♠♦❧❞á✈❡❧✳

❯♠❛ ✈❛r✐❡❞❛❞❡ ❜✐❞✐♠❡♥s✐♦♥❛❧ é ✉♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ ❝♦♠ ❛s ♠❡s♠❛s ♣r♦♣r✐❡❞❛❞❡s ❧♦❝❛✐s ❞♦ ♣❧❛♥♦ ❡✉❝❧✐❞✐❛♥♦✳ ❊ ❛ ❣❡♥❡r❛❧✐③❛çã♦ ♣❛r❛ ❞✐♠❡♥sõ❡s s✉♣❡r✐♦r❡s sã♦ ❛s ✈❛r✐❡✲ ❞❛❞❡s ❞❡ ❞✐♠❡♥sã♦ n✱ q✉❡ sã♦ ❡s♣❛ç♦s t♦♣♦❧ó❣✐❝♦s ❝♦♠ ❛s ♠❡s♠❛s ♣r♦♣r✐❡❞❛❞❡s ❧♦❝❛✐s

❞♦ ❡s♣❛ç♦ ❡✉❝❧✐❞✐❛♥♦ ❞❡ ❞✐♠❡♥sã♦ n✳

❆ ❢♦r♠❛❧✐③❛çã♦ ❞♦ ❝♦♥❝❡✐t♦ ❞❡ ✈❛r✐❡❞❛❞❡ t♦♣♦❧ó❣✐❝❛ ❞❡ ❞✐♠❡♥sã♦ n é ❛♣r❡s❡♥t❛❞❛

♥♦ ❈❛♣ít✉❧♦ ✸✳ ◆♦s ❞❡♠❛✐s ❝❛♣ít✉❧♦s✱ ♥♦s ❡s♣❡❝✐❛❧✐③❛♠♦s ♥♦ ❝❛s♦ n = 2✱ ❝♦♠ ❡s♣❡❝✐❛❧

❛t❡♥çã♦ ❛♦ t❡♦r❡♠❛ ❞❡ ❝❧❛ss✐✜❝❛çã♦ ❞❛s s✉♣❡r❢í❝✐❡s ❝♦♠♣❛❝t❛s✳ ❈♦♠♦ ❝♦♥s❡q✉ê♥❝✐❛ ❞❡ss❡ r❡s✉❧t❛❞♦✱ ♠❛✐s ♦ ❝♦♥❝❡✐t♦ ❞❡ ❝❛r❛❝t❡ríst✐❝❛ ❞❡ ❊✉❧❡r ❞❡ ✉♠❛ s✉♣❡r❢í❝✐❡✱ ❝❤❡❣❛✲s❡ ❛ ✉♠ ♣r♦❝❡❞✐♠❡♥t♦ ❛❧❣é❜r✐❝♦ s✐♠♣❧❡s ♣❛r❛ ❞❡❝✐❞✐r q✉❛♥❞♦ ❞✉❛s s✉♣❡r❢í❝✐❡s ❝♦♠♣❛❝t❛s ❞❛❞❛s sã♦ ❤♦♠❡♦♠♦r❢❛s ♦✉ ♥ã♦✳ ❚❛❧ s✐t✉❛çã♦ ♥ã♦ ♦❝♦rr❡ ❡♠ ❞✐♠❡♥sõ❡s s✉♣❡r✐♦r❡s✳

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✷ Pr❡❧✐♠✐♥❛r❡s

❊ss❡ ❝❛♣ít✉❧♦ é ❜❛s❡❛❞♦ ♥❛ r❡❢❡rê♥❝✐❛ ❬✹❪✱ ❡ t❡♠ ♦ ✐♥t✉✐t♦ ❞❡ ❡st❛❜❡❧❡❝❡r ❛ ♥♦♠❡♥✲ ❝❧❛t✉r❛✱ ❞❡✜♥✐çõ❡s ❡ r❡s✉❧t❛❞♦s ❜ás✐❝♦s ❞❛ ❚♦♣♦❧♦❣✐❛ ●❡r❛❧ q✉❡ s❡rã♦ ✉t✐❧✐③❛❞♦s ♥♦s ❝❛♣ít✉❧♦s s❡❣✉✐♥t❡s✳

❉❡✜♥✐çã♦ ✷✳✶✳ ❙❡❥❛ A ✉♠ ❝♦♥❥✉♥t♦✳ ❉✐③❡♠♦s q✉❡ ✉♠❛ ❢✉♥çã♦ d:A×AR é ✉♠❛

♠étr✐❝❛ s♦❜r❡ A s❡ s❛t✐s❢❛③✿

✶✳ d(a, b)0,a, bA ❡ d(a, b) = 0a=b;

✷✳ d(a, b) =d(b, a),a, bA;

✸✳ d(a, c)≤d(a, b) +d(b, c),∀a, b, c∈A.

◆❡ss❛s ❝♦♥❞✐çõ❡s✱ ♦ ♣❛r (A, d) é ❝❤❛♠❛❞♦ ❡s♣❛ç♦ ♠étr✐❝♦✳

❉❡✜♥✐çã♦ ✷✳✷✳ ❙❡❥❛♠ (A, dA) ❡ (B, dB) ❡s♣❛ç♦s ♠étr✐❝♦s✳ ❯♠❛ ❢✉♥çã♦ f : A → B✱ ❡♥tr❡ ♦s ❡s♣❛ç♦s ♠étr✐❝♦s (A, dA)❡ (B, dB)✱ é ❞✐t❛ s❡r ✉♠❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛ ❡♠x0 ∈A s❡✿ ∀ε >0,∃δ >0 t❛❧ q✉❡✱ dA(x, x0)< δ ⇒dB(f(x), f(x0))< ε.

❙❡ f é ❝♦♥tí♥✉❛ ❡♠ t♦❞♦x0 ∈A✱ ❞✐③❡♠♦s q✉❡f é ❝♦♥tí♥✉❛✳

❉❡✜♥✐çã♦ ✷✳✸✳ ❙❡❥❛ (A, dA) ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦✳ ❙❡❥❛♠x0 ∈A ❡ r >0✳ ❆ ❜♦❧❛ ❛❜❡rt❛ ❞❡ ❝❡♥tr♦ x0 ❡ r❛✐♦ r >0✱ Br(x0)✱ é ❞❡✜♥✐❞❛ ♣♦r

Br(x0) = {x∈A|dA(x, x0)< r}.

❉❡✜♥✐çã♦ ✷✳✹✳ ❙❡❥❛ (A, dA) ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦ ❡ s❡❥❛ U ⊂ A✳ ❉✐③❡♠♦s q✉❡ U é ✉♠ s✉❜❝♦♥❥✉♥t♦ ❛❜❡rt♦ ❞♦ ❡s♣❛ç♦ ♠étr✐❝♦ (A, dA) s❡✿

∀xU,rx >0 t❛❧ q✉❡ Brx(x)⊂U

(23)

✷✷ Pr❡❧✐♠✐♥❛r❡s

❚❡♦r❡♠❛ ✷✳✶✳ ❙❡❥❛♠ (A, dA) ❡ (B, dB) ❡s♣❛ç♦s ♠étr✐❝♦s✳ ❉❛❞❛ f : A → B✱ f é ❝♦♥tí♥✉❛ s❡✱ ❡ s♦♠❡♥t❡ s❡✱f−1(U)é ✉♠ s✉❜❝♦♥❥✉♥t♦ ❛❜❡rt♦ ❞❡A ♣❛r❛ t♦❞♦ s✉❜❝♦♥❥✉♥t♦ ❛❜❡rt♦ U ❞❡ B✳

❉❡♠♦str❛çã♦✿ ❱❡r ❬✹❪✱ ♣á❣✐♥❛ ✾✳

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

✉♠❛ ❢❛♠í❧✐❛ ❞❡ s✉❜❝♦♥❥✉♥t♦s ❞❡ X q✉❡ ♣♦ss✉✐ ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿

✶✳ ∅, X τ;

✷✳ ❙❡ A1, A2, ..., An∈τ ❡♥tã♦ A1∩A2∩...∩An ∈τ; ✸✳ ❙❡ {Ai}iI é ✉♠❛ ❢❛♠í❧✐❛ ❞❡ ❡❧❡♠❡♥t♦s ❞❡ τ ❡♥tã♦

[

i∈I

Ai é ✉♠ ❡❧❡♠❡♥t♦ ❞❡ τ.

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

❉❡✜♥✐çã♦ ✷✳✻✳ ❯♠ s✉❜❝♦♥❥✉♥t♦ C ❞❡ ✉♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ X é ❝❤❛♠❛❞♦ ❝♦♥❥✉♥t♦

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

❉❡✜♥✐çã♦ ✷✳✼✳ ❙❡❥❛ X ✉♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦✳ ❯♠ s✉❜❝♦♥❥✉♥t♦ N ⊆X ❝♦♠ x ∈N é

❝❤❛♠❛❞♦ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❞❡ x s❡ ❡①✐st❡ ✉♠ ❝♦♥❥✉♥t♦ ❛❜❡rt♦ U t❛❧ q✉❡ x∈U ⊆N✳

❉❡✜♥✐çã♦ ✷✳✽✳ ❯♠❛ ❢✉♥çã♦ f :X Y ❡♥tr❡ ❞♦✐s ❡s♣❛ç♦s t♦♣♦❧ó❣✐❝♦s✱ X ❡ Y✱ é ❞✐t❛

s❡r ❝♦♥tí♥✉❛ s❡✱ ♣❛r❛ ❝❛❞❛ ❝♦♥❥✉♥t♦ ❛❜❡rt♦ U ❞❡ Y✱ ❛ ✐♠❛❣❡♠ ✐♥✈❡rs❛ f−1(U) é ❛❜❡rt♦ ❡♠ X✳

❉❡✜♥✐çã♦ ✷✳✾✳ ❯♠❛ ❢✉♥çã♦ q✉❡ ❧❡✈❛ ❝♦♥❥✉♥t♦ ❛❜❡rt♦ ❡♠ ❝♦♥❥✉♥t♦ ❛❜❡rt♦ é ❝❤❛♠❛❞❛ ❢✉♥❝ã♦ ❛❜❡rt❛✳

❉❡✜♥✐çã♦ ✷✳✶✵✳ ❯♠❛ ❢✉♥çã♦ q✉❡ ❧❡✈❛ ❝♦♥❥✉♥t♦ ❢❡❝❤❛❞♦ ❡♠ ❝♦♥❥✉♥t♦ ❢❡❝❤❛❞♦ é ❝❤❛✲ ♠❛❞❛ ❢✉♥çã♦ ❢❡❝❤❛❞❛✳

❚❡♦r❡♠❛ ✷✳✷✳ ❙❡❥❛♠ X, Y ❡ Z ❡s♣❛ç♦s t♦♣♦❧ó❣✐❝♦s✳ ❙❡ f :X Y ❡ g :Y Z sã♦

❢✉♥çõ❡s ❝♦♥tí♥✉❛s ❡♥tã♦ gf :X Z é ❝♦♥tí♥✉❛✳

❉❡♠♦str❛çã♦✿ ❱❡r ❬✹❪✱ ♣á❣✐♥❛ ✶✽✳

❉❡✜♥✐çã♦ ✷✳✶✶✳ ❙❡❥❛♠ X ❡ Y ❡s♣❛ç♦s t♦♣♦❧ó❣✐❝♦s✳ ❉✐③❡♠♦s q✉❡ X ❡ Y sã♦ ❤♦♠❡♦✲

♠♦r❢♦s s❡ ❡①✐st❡♠ ❢✉♥çõ❡s ❝♦♥tí♥✉❛s f : X Y ❡ g : Y X t❛✐s q✉❡ f g = 1Y ❡

(24)

✷✸

❉❡✜♥✐çã♦ ✷✳✶✷✳ ❙❡❥❛ X ✉♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ ❡ s❡❥❛ S ⊂ X✳ ❆ t♦♣♦❧♦❣✐❛ s♦❜r❡ S

✐♥❞✉③✐❞❛ ♣❡❧❛ t♦♣♦❧♦❣✐❛ ❞❡ X é ❛ ❢❛♠í❧✐❛ ❞♦s ❝♦♥❥✉♥t♦s ❞❛ ❢♦r♠❛ U ∩S✱ ♦♥❞❡ U é ✉♠

❝♦♥❥✉♥t♦ ❛❜❡rt♦ ❡♠ X✳

❉❡✜♥✐çã♦ ✷✳✶✸✳ ❙✉♣♦♥❤❛ q✉❡ f : X → Y é ✉♠❛ ❢✉♥çã♦ s♦❜r❡❥❡t✐✈❛ ❞❡ ✉♠ ❡s♣❛ç♦

t♦♣♦❧ó❣✐❝♦ X s♦❜r❡ ✉♠ ❝♦♥❥✉♥t♦ Y✳ ❆ t♦♣♦❧♦❣✐❛ q✉♦❝✐❡♥t❡ s♦❜r❡ Y ❝♦♠ r❡s♣❡✐t♦ ❛ f é

❛ t♦♣♦❧♦❣✐❛ q✉❡ t❡♠ ❝♦♠♦ ❛❜❡rt♦s ❛ ❢❛♠í❧✐❛

τf ={U ⊂Y |f−1(U) é ❛❜❡rt♦ ❡♠ X}.

❚❡♦r❡♠❛ ✷✳✸✳ ❙❡❥❛ f : X Y ✉♠❛ ❢✉♥çã♦ s♦❜r❡❥❡t✐✈❛ ❡ s✉♣♦♥❤❛ q✉❡ Y ♣♦ss✉✐ ❛

t♦♣♦❧♦❣✐❛ q✉♦❝✐❡♥t❡ ❝♦♠ r❡s♣❡✐t♦ ❛X✳ ❊♥tã♦✱ ✉♠❛ ❢✉♥çã♦ g :Y Z ❞❡ Y ♥✉♠ ❡s♣❛ç♦

t♦♣♦❧ó❣✐❝♦ Z é ❝♦♥tí♥✉❛ s❡✱ ❡ s♦♠❡♥t❡ s❡ gf :X Z é ❝♦♥tí♥✉❛✳

❉❡♠♦str❛çã♦✿ ❱❡r ❬✹❪✱ ♣á❣✐♥❛ ✷✽✳

❉❡✜♥✐çã♦ ✷✳✶✹✳ ❙❡❥❛♠ (X, τX) ❡ (Y, τY) ❡s♣❛ç♦s t♦♣♦❧ó❣✐❝♦s ❡ ❝♦♥s✐❞❡r❡ ♦ ♣r♦❞✉t♦ ❝❛rt❡s✐❛♥♦

X×Y ={(x, y)|xX ❡ yY}.

❆ t♦♣♦❧♦❣✐❛ ♣r♦❞✉t♦ ♥❡ss❡ ❝♦♥❥✉♥t♦ ❝♦♥s✐st❡ ❞♦ s❡❣✉✐♥t❡✿ ♦s ❝♦♥❥✉♥t♦s ❛❜❡rt♦s sã♦ t♦❞♦s ♦s ❝♦♥❥✉♥t♦s ❞❛ ❢♦r♠❛ [

i∈I

Ai×Bi✱ ♦♥❞❡Ai é ❛❜❡rt♦ ❡♠ X ❡ Bi é ❛❜❡rt♦ ❡♠ Y✳ ❚❡♦r❡♠❛ ✷✳✹✳ ❙❡❥❛♠ A, X ❡ Y ❡s♣❛ç♦s t♦♣♦❧ó❣✐❝♦s ❡ s❡❥❛♠ f : A X ❡ g : A Y

❢✉♥çõ❡s✳ ❉❡✜♥❛ h= (f, g) :A X×Y ♣♦r h(a) = (f(a), g(a))✳ ❊♥tã♦✱ h é ❝♦♥tí♥✉❛

s❡✱ ❡ s♦♠❡♥t❡ s❡✱ f ❡ g sã♦ ❝♦♥tí♥✉❛s✳

❉❡♠♦♥str❛çã♦✿ ❱❡r ❬✹❪✱ ♣á❣✐♥❛ ✹✶✳

❉❡✜♥✐çã♦ ✷✳✶✺✳ ❉❛❞♦ ✉♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ X ❡ ✉♠ s✉❜❝♦♥❥✉♥t♦ S ❞❡ X✱ ✉♠❛ ❝♦❜❡r✲

t✉r❛ ❞❡ S é ✉♠❛ ❢❛♠í❧✐❛ {Uj}j∈J ❞❡ s✉❜❝♦♥❥✉♥t♦s ❞❡ X t❛❧ q✉❡ S ⊆

[

j∈J

Uj✳ ◆♦ ❝❛s♦ q✉❡ J é ✜♥✐t♦✱ ❞✐③❡♠♦s q✉❡ ❡ss❛ ❝♦❜❡rt✉r❛ é ✜♥✐t❛✳

❉❡✜♥✐çã♦ ✷✳✶✻✳ ❉❛❞❛s ❝♦❜❡rt✉r❛s {Uj}j∈J ❡ {Vk}k∈K✱ ❞✐③❡♠♦s q✉❡ {Uj}j∈J é ✉♠❛ s✉❜❝♦❜❡rt✉r❛ ❞❡ {Vk}k∈K s❡ ♣❛r❛ t♦❞♦ j ∈J ❡①✐st❡ ✉♠ k ∈K t❛❧ q✉❡ Uj =Vk✳

❉❡✜♥✐çã♦ ✷✳✶✼✳ ❉❛❞♦sX ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦✱S X ❡{Uj}j∈J ❝♦❜❡rt✉r❛ ❞❡S✱ ❞✐③❡♠♦s q✉❡ ❡ss❛ é ✉♠❛ ❝♦❜❡rt✉r❛ ❛❜❡rt❛ ❞❡ S s❡ Uj é ❛❜❡rt♦ ❡♠ X✱ ∀j ∈J✳

❉❡✜♥✐çã♦ ✷✳✶✽✳ ❯♠ s✉❜❝♦♥❥✉♥t♦S ❞❡X é ❝❤❛♠❛❞♦ ❝♦♠♣❛❝t♦ s❡ t♦❞❛ ❝♦❜❡rt✉r❛ ❛❜❡rt❛

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✷✹ Pr❡❧✐♠✐♥❛r❡s

❚❡♦r❡♠❛ ✷✳✺✳ ❚♦❞♦ ✐♥t❡r✈❛❧♦ ❞❛ ❢♦r♠❛ [a, b]✱ ❝♦♠ a, b∈R ❡ a < b✱ é ✉♠ s✉❜❝♦♥❥✉♥t♦

❝♦♠♣❛❝t♦ ❞❡ R✳

❉❡♠♦♥str❛çã♦✿ ❱❡r ❬✹❪✱ ♣á❣✐♥❛ ✹✻✳

❚❡♦r❡♠❛ ✷✳✻✳ ❙❡❥❛f :X →Y ✉♠❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛✳ ❙❡S ⊂Xé ✉♠ ❡s♣❛ç♦ ❝♦♠♣❛❝t♦

❡♥tã♦ f(S) é ❝♦♠♣❛❝t♦ ❡♠ Y✳

❉❡♠♦♥str❛çã♦✿ ❱❡r ❬✹❪✱ ♣á❣✐♥❛ ✹✻✳

❚❡♦r❡♠❛ ✷✳✼✳ ❯♠ s✉❜❝♦♥❥✉♥t♦ ❢❡❝❤❛❞♦ ❞❡ ✉♠ ❡s♣❛ç♦ ❝♦♠♣❛❝t♦ é ❝♦♠♣❛❝t♦✳ ❉❡♠♦♥str❛çã♦✿ ❱❡r ❬✹❪✱ ♣á❣✐♥❛ ✹✼✳

❚❡♦r❡♠❛ ✷✳✽✳ ❙❡❥❛♠ X ❡ Y ❡s♣❛ç♦s t♦♣♦❧ó❣✐❝♦s✳ ❊♥tã♦✱ X ❡ Y sã♦ ❝♦♠♣❛❝t♦s s❡✱ ❡

s♦♠❡♥t❡ s❡✱ X×Y é ❝♦♠♣❛❝t♦✳

❉❡♠♦♥str❛çã♦✿ ❱❡r ❬✹❪✱ ♣á❣✐♥❛ ✹✼✳

❉❡✜♥✐çã♦ ✷✳✶✾✳ ❯♠ s✉❜❝♦♥❥✉♥t♦ S ❞❡ Rn é ❞✐t♦ ❧✐♠✐t❛❞♦ s❡ ❡①✐st❡ ✉♠ ♥ú♠❡r♦ r❡❛❧

k > 0 t❛❧ q✉❡ ♣❛r❛ ❝❛❞❛ ♣♦♥t♦ x = (x1, ..., xn) ∈ S✱ | xi |≤ k ♣❛r❛ i = 1,2, ..., n✳ ❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ S ✧✈✐✈❡✧ ❞❡♥tr♦ ❞♦ ❝✉❜♦ ❞❡ ❞✐♠❡♥sã♦ n✱ [k, k]×...×[k, k]✳

❚❡♦r❡♠❛ ✷✳✾✳ ❯♠ s✉❜❝♦♥❥✉♥t♦ ❢❡❝❤❛❞♦ ❡ ❧✐♠✐t❛❞♦ ❞❡ Rn é ❝♦♠♣❛❝t♦✳

❉❡♠♦♥str❛çã♦✿ ❱❡r ❬✹❪✱ ♣á❣✐♥❛ ✹✽✳

❉❡✜♥✐çã♦ ✷✳✷✵✳ ❯♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ X é ✉♠ ❡s♣❛ç♦ ❞❡ ❍❛✉s❞♦r✛ s❡ ♣❛r❛ t♦❞♦ x, y ∈X, x6=y✱ ❡①✐st❡♠ ❛❜❡rt♦s Ux✱ Uy ❞❡ X t❛✐s q✉❡ x∈Ux✱ y∈Uy ❡ Ux∩Uy =∅. ❚❡♦r❡♠❛ ✷✳✶✵✳ ❙❡ X é ✉♠ ❡s♣❛ç♦ ❞❡ ❍❛✉s❞♦r✛ ❡ Y X é ✉♠ ❡s♣❛ç♦ ❝♦♠♣❛❝t♦✱

❡♥tã♦ Y é ❢❡❝❤❛❞♦ ❡♠ X✳

❉❡♠♦♥str❛çã♦✿ ❱❡r ❬✹❪✱ ♣á❣✐♥❛ ✺✷✳

❚❡♦r❡♠❛ ✷✳✶✶✳ ❙✉♣♦♥❤❛f :X Y ✉♠❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛✱ ❝♦♠ X ❡s♣❛ç♦ ❝♦♠♣❛❝t♦ ❡ Y ❡s♣❛ç♦ ❞❡ ❍❛✉s❞♦r✛✳ ❊♥tã♦✱ f é ✉♠ ❤♦♠❡♦♠♦r✜s♠♦ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ f é ❜✐❥❡t♦r❛✳

❉❡♠♦♥str❛çã♦✿ ❱❡r ❬✹❪✱ ♣á❣✐♥❛ ✺✷✳

❉❡✜♥✐çã♦ ✷✳✷✶✳ ❯♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ X é ✉♠ ❡s♣❛ç♦ ❝♦♥❡①♦ s❡ s❡✉s ú♥✐❝♦s s✉❜❝♦♥✲

❥✉♥t♦s s✐♠✉❧t❛♥❡❛♠❡♥t❡ ❛❜❡rt♦ ❡ ❢❡❝❤❛❞♦ sã♦ ❛♣❡♥❛s ♦ ∅ ❡ X✳

❚❡♦r❡♠❛ ✷✳✶✷✳ X é ❝♦♥❡①♦ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ X ♥ã♦ ♣♦❞❡ s❡r ❡s❝r✐t♦ ❝♦♠♦ ❛ r❡✉♥✐ã♦

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✸ ❱❛r✐❡❞❛❞❡s t♦♣♦❧ó❣✐❝❛s

❉❡✜♥✐çã♦ ✸✳✶✳ ❙❡❥❛ n ✉♠ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦✳ ❯♠❛ ✈❛r✐❡❞❛❞❡ t♦♣♦❧ó❣✐❝❛ ❞❡ ❞✐♠❡♥sã♦ n

✭♦✉ n✲✈❛r✐❡❞❛❞❡✮ é ✉♠ ❡s♣❛ç♦ ❞❡ ❍❛✉s❞♦r✛ ♦♥❞❡ ❝❛❞❛ ♣♦♥t♦ ♣♦ss✉✐ ✉♠❛ ✈✐③✐♥❤❛♥ç❛

❤♦♠❡♦♠♦r❢❛ ❛♦ ❞✐s❝♦ ❛❜❡rt♦ ❞❡ ❞✐♠❡♥sã♦ n✱ Un={xRn | |x|<1}✳

❯s❛r❡♠♦s s♦♠❡♥t❡ ♦ t❡r♠♦ ✈❛r✐❡❞❛❞❡ ♣❛r❛ r❡❢❡r❡♥❝✐❛r♠♦s ❛ ✉♠❛ ✈❛r✐❡❞❛❞❡ t♦♣♦❧ó✲ ❣✐❝❛✳

❖❜s❡r✈❛çã♦ ✸✳✶✳ P♦r ❞❡✜♥✐çã♦✱ s❡ U é ❛❜❡rt♦ ❡♠ Rn ❡ xU ❡♥tã♦ ❡①✐st❡ ✉♠❛ ❜♦❧❛

❛❜❡rt❛ ❡♠Rn❝♦♠ ❝❡♥tr♦ ❡♠x❡ q✉❡ ❡stá ❝♦♥t✐❞❛ ❡♠U✳ P♦rt❛♥t♦✱ ♣♦❞❡♠♦s ❡♥❢r❛q✉❡❝❡r

❛ ❝♦♥❞✐çã♦ ♥❛ ❞❡✜♥✐çã♦ ♣♦♥❞♦✿ ✏❝❛❞❛ ♣♦♥t♦ ♣♦ss✉✐ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❤♦♠❡♦♠♦r❢❛ ❛ ✉♠ ❛❜❡rt♦ ❞❡ Rn✳✑

❊①❡♠♣❧♦ ✸✳✶✳ ❖ ❡s♣❛ç♦ ❡✉❝❧✐❞✐❛♥♦ Rn é ♦❜✈✐❛♠❡♥t❡ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞❡ ❞✐♠❡♥sã♦ n ❇❛st❛ ✈❡r✐✜❝❛r q✉❡ ❛ ❢✉♥çã♦ ✐❞❡♥t✐❞❛❞❡ é ✉♠ ❤♦♠❡♦♠♦r✜s♠♦ ❞❡ Rn s♦❜r❡ ✉♠ ❛❜❡rt♦ ❞❡ Rn✱ ❛ s❛❜❡r✱ ❡❧❡ ♠❡s♠♦✳

❊①❡♠♣❧♦ ✸✳✷✳ ❆ ❡s❢❡r❛

Sn ={x= (x1, . . . , xn+1)∈Rn+1 | x21+· · ·+x2n+1 = 1}

é ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞❡ ❞✐♠❡♥sã♦ n✳ P❛r❛ ❛ ♣r♦✈❛ ❞❡ t❛❧ ❢❛t♦✱ ♥♦t❡ ♣r✐♠❡✐r❛♠❡♥t❡ q✉❡

Sn =Sn

\ {P} ∪Sn

\ {Q},

♦♥❞❡ P = (0, . . . ,0,1) ❡ Q = (0, . . . ,0,−1)✳ ✭◆♦ ❝❛s♦ n = 2✱ P ❡ Q sã♦ ♦s ♣ó❧♦s

♥♦rt❡ ❡ s✉❧✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✮✳ Pr♦✈❛r❡♠♦s ❛❣♦r❛ q✉❡ Sn \ {Q} é ❤♦♠❡♦♠♦r❢♦ ❛ Rn ✭❆♥❛❧♦❣❛♠❡♥t❡✱ ♣r♦✈❛✲s❡ q✉❡ Sn\ {P}é ❤♦♠❡♦♠♦r❢♦ ❛Rn✱ ❞❡ ♠♦❞♦ ❛ ❝♦♥❝❧✉✐r♠♦s q✉❡

Sn é ✉♠❛ n✲✈❛r✐❡❞❛❞❡✮✳

❉❡✜♥❛f :Sn\ {Q} →Rn ♣♦r

f(x1, . . . , xn+1) =

x1

1 +xn+1

, . . . , xn

1 +xn+1

.

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✷✻ ❱❛r✐❡❞❛❞❡s t♦♣♦❧ó❣✐❝❛s

❆ ✐♥t❡r♣r❡t❛çã♦ ♣❛r❛ ♦ ♣♦♥t♦ f(x) ∈ Rn é q✉❡ ♦ ♣♦♥t♦ Y = (f(x),0) ∈ Rn+1 é

❡①❛t❛♠❡♥t❡ ♦ ♣♦♥t♦ ❞❡ ✐♥t❡rs❡çã♦ ❡♥tr❡ ❛ r❡t❛ r ❞❡t❡r♠✐♥❛❞❛ ♣❡❧♦s ♣♦♥t♦s x ❡ Q ❡ ♦

❤✐♣❡r❡s♣❛ç♦ xn+1 = 0✳ ❉❡ ❢❛t♦✱ ❞❛❞♦ x ∈ Sn \ {Q}✱ ❝♦♥s✐❞❡r❡ ❛ r❡t❛ r ❞❡t❡r♠✐♥❛❞❛ ♣❡❧♦s ♣♦♥t♦s x ❡ Q✱ ✐✳❡✳✱

r :Y =x+t(Qx), tR.

❚❡♠♦s q✉❡ ❛ r❡t❛ r ❛tr❛✈❡ss❛ ♦ ❤✐♣❡r❡s♣❛ç♦ xn+1 = 0 ♥✉♠ ú♥✐❝♦ ♣♦♥t♦ Y✳ P❛r❛ ❞❡t❡r♠✐♥❛r♠♦s ❛s ❝♦♦r❞❡♥❛❞❛s ❞❡ Y✱ s❡❥❛ x= (x1, . . . , xn+1)✳ ❊♥tã♦

Y = ((1t)x1, . . . , (1−t)xn, xn+1+t(−1−xn+1))

❡ ❞❡✈❡♠♦s ❡♥tã♦ r❡s♦❧✈❡r ❛ ❡q✉❛çã♦ ❡♠t✿

xn+1+t(−1−xn+1) = 0,

❛ q✉❛❧ ♥♦s ❢♦r♥❡❝❡t = xn+1 1 +xn+1✳

❈♦♥❤❡❝❡♥❞♦ ❛❣♦r❛ ♦ ✈❛❧♦r ❞❡ t✱ t❡♠♦s q✉❡ ♦ ♣♦♥t♦ Y ❞❡ ✐♥t❡rs❡çã♦ ❡♥tr❡ ♦ ❤✐♣❡✲

r❡s♣❛ç♦ xn+1 = 0 ❡ ❛ r❡t❛ ❞❡t❡r♠✐♥❛❞❛ ♣❡❧♦s ♣♦♥t♦s x ❡ Q é ❡s❝r✐t♦ ❡♠ ❝♦♦r❞❡♥❛❞❛s ❝❛rt❡s✐❛♥❛s ❝♦♠♦

Y =

x1

1 +xn+1

, . . . , xn

1 +xn+1

,0

.

◆❡ss❡ ♣♦♥t♦✱ ❞❡✜♥✐❞❛ ❛ ❢✉♥çã♦ f ❡ ❡①♣❧✐❝✐t❛❞❛ ❛ s✉❛ ✐♥t❡r♣r❡t❛çã♦✱ ♣r♦✈❛r❡♠♦s q✉❡ f é ✉♠ ❤♦♠❡♦♠♦r✜s♠♦✳ ❆ ✈❡r✐✜❝❛çã♦ ❞❡ q✉❡f é ❝♦♥tí♥✉❛ é ✐♠❡❞✐❛t❛✱ ♣♦✐s ❝❛❞❛ ✉♠❛ ❞❡

s✉❛s ❢✉♥çõ❡s ❝♦♦r❞❡♥❛❞❛s ♦ sã♦✳ P❛r❛ ❝♦♥❝❧✉✐r♠♦s q✉❡f é ❤♦♠❡♦♠♦r✜s♠♦✱ ❞❡✜♥✐r❡♠♦s

✉♠❛ ❢✉♥çã♦ g : Rn Sn

\ {Q} ❞❡ t❛❧ s♦rt❡ q✉❡ g s❡❥❛ ❝♦♥tí♥✉❛✱ g f = idSn\{Q}

f g =idRn✳ ❆ ❢✉♥çã♦ g é ❞❡✜♥✐❞❛ ♣♦r✿

g(a) =

2a1

a2

1+· · ·+a2n+ 1

, . . . , 2an a2

1+· · ·+a2n+ 1

,1−(a

2

1+· · ·+a2n)

a2

1+· · ·+a2n+ 1

,

♦♥❞❡ a = (a1, . . . , an) ∈ Rn✳ ❆ ❢✉♥çã♦ g é ❝♦♥tí♥✉❛ ♣♦✐s ❝❛❞❛ ✉♠❛ ❞❡ s✉❛s ❢✉♥çõ❡s ❝♦♦r❞❡♥❛❞❛s ♦ sã♦✳ ❆❞❡♠❛✐s✱ ✈❡r✐✜❝❛✲s❡ q✉❡ g ❡stá ❜❡♠ ❞❡✜♥✐❞❛✱ ✐✳❡✳✱ q✉❡ g(a) Sn ♣❛r❛ t♦❞♦ a Rn✳ ❚❛♠❜é♠ ♥ã♦ é ❞✐❢í❝✐❧ ✈❡r✐✜❝❛r ❛s ✐❣✉❛❧❞❛❞❡s g f = id

Sn

\{Q} ❡

f g =idRn✳ ▲♦❣♦✱ f é ✉♠ ❤♦♠❡♦♠♦r✜s♠♦✳

P♦rt❛♥t♦✱ ❛ ❡s❢❡r❛ Sn é ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞❡ ❞✐♠❡♥sã♦n

❊①❡♠♣❧♦ ✸✳✸✳ ❙❡ Mn é ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞❡ ❞✐♠❡♥sã♦ n✱ ❡♥tã♦ q✉❛❧q✉❡r s✉❜❝♦♥❥✉♥t♦ ❛❜❡rt♦ ❞❡ Mn é t❛♠❜é♠ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞❡ ❞✐♠❡♥sã♦ n✳ ❆ ♣r♦✈❛ é ✐♠❡❞✐❛t❛✳

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✷✼

q✉❡ ❞❛❞♦ ✉♠ ♣♦♥t♦z ∈M×N✱ ❡①✐st❡ ✉♠ ❛❜❡rt♦Up×Uq⊂M×N ❤♦♠❡♦♠♦r❢♦ ❛ ✉♠ ❛❜❡rt♦ ❡♠ Rm+n✳ ❙❡❥❛ z = (p, q) ∈ M ×N✳ ❈♦♠♦ M é ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞❡ ❞✐♠❡♥sã♦ m✱ ❡①✐st❡ ✉♠ ❤♦♠❡♦♠♦r✜s♠♦Ψ1 :Up →Rm✱ s❡♥❞♦ Up ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❛❜❡rt❛ ❞❡ p❡♠

M✳ ❆♥❛❧♦❣❛♠❡♥t❡✱ ❡①✐st❡ ✉♠ ❤♦♠❡♦♠♦r✜s♠♦Ψ2 :Uq→Rn✱ s❡♥❞♦Uq ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❛❜❡rt❛ ❞❡ q ❡♠ N✳

❖ ❝♦♥❥✉♥t♦Up×Uq é ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❛❜❡rt❛ ❞❡z = (p, q)❡♠ M×N ❡ ❛ ❢✉♥çã♦f :

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✹ ❱❛r✐❡❞❛❞❡s ❜✐❞✐♠❡♥s✐♦♥❛✐s ✲

♦r✐❡♥tá✈❡✐s ❡ ♥ã♦✲♦r✐❡♥tá✈❡✐s✳

❯♠❛ ❞❡✜♥✐çã♦ ❢♦r♠❛❧ ❞❡ ♦r✐❡♥t❛çã♦ ❡♠ ✈❛r✐❡❞❛❞❡s ❜✐❞✐♠❡♥s✐♦♥❛✐s ♥ã♦ ❡stá ♥♦ ♥♦ss♦ ❛❧❝❛♥❝❡ ♥❡ss❡ ♠♦♠❡♥t♦✳ ❆ss✐♠✱ ♥♦s r❡♥❞❡r❡♠♦s ❛ ✉♠ tr❛t❛♠❡♥t♦ ✐♥t✉✐t✐✈♦✳ P❛r❛ ❞❡✜♥✐r ♦ ❝♦♥❝❡✐t♦ ❞❡ ✈❛r✐❡❞❛❞❡ ❜✐❞✐♠❡♥s✐♦♥❛❧ ♥ã♦✲♦r✐❡♥tá✈❡❧✱ ❝♦♠❡ç❛r❡♠♦s ❝♦♠ ♦ ❡①❡♠♣❧♦ ❝❧áss✐❝♦✿ ❛ ❢❛✐①❛ ❞❡ ▼ö❜✐✉s✳

❆ ❢❛✐①❛ ❞❡ ▼ö❜✐✉s é ❝♦♥str✉í❞❛ ❛ ♣❛rt✐r ❞❡ ✉♠❛ t✐r❛ ❞❡ ♣❛♣❡❧ ❝♦❧❛♥❞♦✲s❡ s✉❛s ❡①tr❡♠✐❞❛❞❡s ❞❡♣♦✐s ❞❡ r❡❛❧✐③❛r ✉♠❛ t♦rçã♦ ❞❡ 180♦ ❡♠ ✉♠❛ ❞❡❧❛s✳ ❈♦♠♦ ❡s♣❛ç♦

t♦♣♦❧ó❣✐❝♦✱ ❡❧❛ é ♦❜t✐❞❛ ♣♦r t♦♣♦❧♦❣✐❛ q✉♦❝✐❡♥t❡ ❞♦ s❡❣✉✐♥t❡ ♠♦❞♦✿ ❙❡❥❛X ♦ r❡tâ♥❣✉❧♦

X ={(x, y)R2 | −10x10,1< y <1}.

❋♦r♠❛♠♦s ❛❣♦r❛ ✉♠ ❡s♣❛ç♦ q✉♦❝✐❡♥t❡ ❞❡X ✐❞❡♥t✐✜❝❛♥❞♦ ♦s ♣♦♥t♦s(10, y)❡(10,y)

♣❛r❛ ❝❛❞❛ y ♥♦ ✐♥t❡r✈❛❧♦ ❛❜❡rt♦ (−1,1)✳

❖❜s❡r✈❡ q✉❡ ❛ ❧✐♥❤❛ ❝❡♥tr❛❧ ❞❛ ❢❛✐①❛ r❡t❛♥❣✉❧❛rX t♦r♥❛✲s❡ ✉♠ ❝ír❝✉❧♦ ❛♣ós ❛ ✐❞❡♥✲

t✐✜❝❛çã♦ ❞❛s ❞✉❛s ❡①tr❡♠✐❞❛❞❡s✳ ❯♠ s❡r ✐♠❛❣✐♥ár✐♦ q✉❡ ❝♦♠❡ç♦✉ ✉♠ ♣❡r❝✉rs♦ ❡♠ q✉❛❧q✉❡r ♣♦♥t♦ ♥❡st❡ ❝ír❝✉❧♦ ❝♦♠ ✉♠❛ ❡s❝♦❧❤❛ ❞❡ ♦r✐❡♥t❛çã♦ ❞❡✜♥✐❞❛ ❡ ❝❛rr❡❣♦✉ ❡st❛ ♦r✐❡♥t❛çã♦ ❝♦♠ ❡❧❡ ❡♠ t♦r♥♦ ❞♦ ❝ír❝✉❧♦✱ ✈♦❧t❛ ♣❛r❛ s❡✉ ♣♦♥t♦ ✐♥✐❝✐❛❧ ❝♦♠ s✉❛ ♦r✐❡♥t❛çã♦ ✐♥✈❡rt✐❞❛✳ ❊st❡ t✐♣♦ ❞❡ ♣❡r❝✉rs♦ ❢❡❝❤❛❞♦ r❡❝❡❜❡ ❛ ❝❧❛ss✐✜❝❛çã♦ ❞❡ ✐♥✈❡rs♦r ❞❡ ♦r✐❡♥✲ t❛çã♦✳ ❯♠ ♣❡r❝✉rs♦ q✉❡ ♥ã♦ ♣♦ss✉❛ ❡ss❛ ♣r♦♣r✐❡❞❛❞❡ é ❞❡♥♦♠✐♥❛❞♦ ♣r❡s❡r✈❛❞♦r ❞❡ ♦r✐❡♥t❛çã♦✳

(31)

✸✵ ❱❛r✐❡❞❛❞❡s ❜✐❞✐♠❡♥s✐♦♥❛✐s ✲ ♦r✐❡♥tá✈❡✐s ❡ ♥ã♦✲♦r✐❡♥tá✈❡✐s✳

❋✐❣✉r❛ ✹✳✶✿ ❋❛✐①❛ ❞❡ ▼ö❜✐✉s✳ ❆ ✜❣✉r❛ ✹✳✶ ❢♦✐ r❡t✐r❛❞❛ ❞❡ ❬✶❪✱ ♣á❣✐♥❛ ✶✼✳

❉❡ ♠♦❞♦ ❣❡r❛❧✱ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❜✐❞✐♠❡♥s✐♦♥❛❧ q✉❡ ❝♦♥t❡♥❤❛ ✉♠ ♣❡r❝✉rs♦ ❢❡❝❤❛❞♦ ✐♥✈❡rs♦r ❞❡ ♦r✐❡♥t❛çã♦ é ❞❡♥♦♠✐♥❛❞❛ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❜✐❞✐♠❡♥s✐♦♥❛❧ ♥ã♦✲♦r✐❡♥tá✈❡❧✳ ❈❛s♦ ❝♦♥trár✐♦✱ é ❝❤❛♠❛❞❛ ♦r✐❡♥tá✈❡❧✳

(32)

✺ ❊①❡♠♣❧♦s ❞❡ s✉♣❡r❢í❝✐❡s ❝♦♠♣❛❝t❛s✳

❈❤❛♠❛r❡♠♦s ✉♠❛2✲✈❛r✐❡❞❛❞❡ ❝♦♥❡①❛ ❛❜r❡✈✐❛❞❛♠❡♥t❡ ❞❡ s✉♣❡r❢í❝✐❡✳

❯♠ ❡①❡♠♣❧♦ ❞❡ ✉♠❛ s✉♣❡r❢í❝✐❡ ❝♦♠♣❛❝t❛ é ❛ 2✲❡s❢❡r❛ S2✳ ❆ ❡s❢❡r❛ S2 é ♦r✐❡♥tá✈❡❧✳ ❖✉tr♦ ❡①❡♠♣❧♦ é ♦2✲t♦r♦ T2✳ ❖ t♦r♦ T2 ♣♦❞❡ s❡r ❞❡s❝r✐t♦ ❝♦♠♦ s❡♥❞♦ ❛ s✉♣❡r❢í❝✐❡ ❞❡ ✉♠❛ ❝â♠❛r❛ ❞❡ ♣♥❡✉ ❝❤❡✐❛ ❞❡ ❛r✳ ❊ ♣♦❞❡ s❡r ❞❡✜♥✐❞♦ ❝♦♠ ♦ r✐❣♦r ♠❛t❡♠át✐❝♦ ❞❡ ✉♠❛ ❞❛s s❡❣✉✐♥t❡s ♠❛♥❡✐r❛s✿

✭❛✮ ◗✉❛❧q✉❡r ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ ❤♦♠❡♦♠♦r❢♦ ❛♦ ♣r♦❞✉t♦ ❞❡ ❞♦✐s ❝ír❝✉❧♦s✱ S1 ×S1. ✭❜✮ ◗✉❛❧q✉❡r ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ ❤♦♠❡♦♠♦r❢♦ ❛♦ s❡❣✉✐♥t❡ s✉❜❝♦♥❥✉♥t♦ ❞♦R3✿

n

(x, y, z)R3 |(x2+y2 2)2+z2 = 1o

✭❡st❡ ❝♦♥❥✉♥t♦ é ♦❜t✐❞♦ ♣❡❧❛ r♦t❛çã♦ ❞♦ ❝ír❝✉❧♦ (x2)2+z2 = 1✱ s✐t✉❛❞♦ ♥♦ ♣❧❛♥♦ xz ❡♠ t♦r♥♦ ❞♦ ❡✐①♦ z✮✳

✭❝✮ ❙❡❥❛X ♦ q✉❛❞r❛❞♦ ✉♥✐tár✐♦ ♥♦ ♣❧❛♥♦ R2

{(x, y)R2 | 0x1,0y1}.

❯♠ t♦r♦ é q✉❛❧q✉❡r ❡s♣❛ç♦ ❤♦♠❡♦♠♦r❢♦ ❛♦ ❡s♣❛ç♦ q✉♦❝✐❡♥t❡ ❞❡ X ♦❜t✐❞♦ ♣❡❧❛ ✐❞❡♥t✐✲

✜❝❛çã♦ ❞♦s ❧❛❞♦s ♦♣♦st♦s ❞♦ q✉❛❞r❛❞♦ X ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛s s❡❣✉✐♥t❡s r❡❣r❛s✿

✶✳ P❛r❛ ❝❛❞❛ y ♥♦ ✐♥t❡r✈❛❧♦[0,1]✱ ♦s ♣♦♥t♦s (0, y) ❡ (1, y) s❡ ✐❞❡♥t✐✜❝❛♠✳

✷✳ P❛r❛ ❝❛❞❛ x ♥♦ ✐♥t❡r✈❛❧♦[0,1]✱ ♦s ♣♦♥t♦s (x,0)❡ (x,1)s❡ ✐❞❡♥t✐✜❝❛♠✳

❊ss❛s ✐❞❡♥t✐✜❝❛çõ❡s ❡stã♦ r❡♣r❡s❡♥t❛❞❛s ♥♦ ❞✐❛❣r❛♠❛ ❛❜❛✐①♦✿

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✸✷ ❊①❡♠♣❧♦s ❞❡ s✉♣❡r❢í❝✐❡s ❝♦♠♣❛❝t❛s✳

❋✐❣✉r❛ ✺✳✶✿ ❚♦r♦✳

▲❛❞♦s ✐❞❡♥t✐✜❝❛❞♦s sã♦ r♦t✉❧❛❞♦s ❝♦♠ ❛ ♠❡s♠❛ ❧❡tr❛ ❞♦ ❛❧❢❛❜❡t♦✱ ❡ ❛s ✐❞❡♥t✐✜❝❛çõ❡s ❞❡✈❡♠ s❡r ❢❡✐t❛s s❡❣✉♥❞♦ ❛s s❡t❛s✳

Pr♦✈❛r❡♠♦s ❛❣♦r❛ q✉❡ ♦s ❡s♣❛ç♦s t♦♣♦❧ó❣✐❝♦s ❞❡s❝r✐t♦s ❡♠ ✭❛✮✱ ✭❜✮ ❡ ✭❝✮ sã♦ ❤♦♠❡✲ ♦♠♦r❢♦s✳

Pr✐♠❡✐r♦✱ t♦♠❡ I = [0,1] ❡ ❝♦♥s✐❞❡r❡ ♦ ❡s♣❛ç♦ q✉♦❝✐❡♥t❡ I×I

∼ ✱ ♦♥❞❡ ∼ ❞❡♥♦t❛ ❛ ✐❞❡♥t✐✜❝❛çã♦ ❞❛❞❛ ♥♦ ✐t❡♠ ✭❝✮✳ ❉❡✜♥✐♠♦s ✉♠ ❤♦♠❡♦♠♦r✜s♠♦

g : I×I

∼ →

n

(x, y, z)R3 |(x2+y22)2+z2 = 1o

♣♦r

g((u, v)) = (p2 + cos(2πu) cos(2πv),p2 + cos(2πu) sen(2πv),sen(2πu)).

P♦❞❡✲s❡ ♣r♦✈❛r q✉❡ ❛ ❢✉♥çã♦g ❡stá ❜❡♠ ❞❡✜♥✐❞❛✳ P❛r❛ ✐ss♦✱ ❞❡✈❡✲s❡ ♣r♦✈❛r q✉❡g((u, v))

✐♥❞❡♣❡♥❞❡ ❞♦ r❡♣r❡s❡♥t❛♥t❡(u, v)I×I ❞❛ ❝❧❛ss❡(u, v)✱ ♦ q✉❡ ♦❜✈✐❛♠❡♥t❡ é ✈❡r❞❛❞❡✱

❡ q✉❡ g((u, v)) ♣❡rt❡♥❝❡ ❞❡ ❢❛t♦ ❛♦ s✉❜❝♦♥❥✉♥t♦

n

(x, y, z)R3 |(x2+y22)2+z2 = 1o

❞❡R3✳ P❛r❛ ❡ss❛ ú❧t✐♠❛ ❛✜r♠❛çã♦✱ t❡♠♦s q✉❡ ✈❡r✐✜❝❛r ❛ s❡❣✉✐♥t❡ ✐❣✉❛❧❞❛❞❡✿

(p2 + cos(2πu)2cos2(2πv) +p2 + cos(2πu)2sen2(2πv)2)2+ sen2(2πu) = 1

❉❡ ❢❛t♦✱

(p2 + cos(2πu)2cos2(2πv) +p2 + cos(2πu)2sen2(2πv)2)2+ sen2(2πu) =

= ((2 + cos(2πu)) cos2(2πv) + (2 + cos(2πu)) sen2(2πv)2)2+ sen2(2πu) =

(34)

✸✸

= (2 +cos(2πu)−2)2+sen2(2πu) =cos2(2πu) +sen2(2πu) = 1

Pr♦✈❛✲s❡ t❛♠❜é♠ q✉❡g é ✐♥❥❡t♦r❛✳ ❉❡ ❢❛t♦✱ s❡❥❛♠ (u, v),(u′, v) I ×I

∼ t❛✐s q✉❡

g((u, v)) =g((u′, v)).

❊♥tã♦✱

(p2 +cos(2πu)cos(2πv),p2 +cos(2πu)sen(2πv), sen(2πu)) = = (p2 +cos(2πu′

)cos(2πv′),p2 +cos(2πu′

)sen(2πv′), sen(2πu′)).

❖❧❤❛♥❞♦ ♣❛r❛ ❛ ✐❣✉❛❧❞❛❞❡ ❞❛s ❞✉❛s ♣r✐♠❡✐r❛s ❝♦♦r❞❡♥❛❞❛s t❡♠♦s✿

  

p

2 +cos(2πu)cos(2πv) =p2 +cos(2πu′

)cos(2πv′ ),

p

2 +cos(2πu)sen(2πv) = p2 +cos(2πu′

)sen(2πv′ ).

❘❡s♦❧✈❡♥❞♦ ♦ s✐st❡♠❛ t❡♠♦s q✉❡✿

[p2 +cos(2πu)cos(2πv)]2 + [p2 +cos(2πu)sen(2πv)]2 = = [p2 +cos(2πu′

)cos(2πv′

)]2+ [p2 +cos(2πu′

)sen(2πv′ )]2

⇒[2 +cos(2πu)][cos2(2πv) +sen2(2πv)] = [2 +cos(2πu

)][cos2(2πv

) +sen2(2πv′ )]⇒ ⇒2 +cos(2πu) = 2 +cos(2πu′)⇒cos(2πu) = cos(2πu′).

■ss♦✱ ✉♥✐❞♦ ❛ ✐❣✉❛❧❞❛❞❡ ❞❛s ú❧t✐♠❛s ❝♦♦r❞❡♥❛❞❛s✱ sen(2πu) = sen(2πu′

)✱ ♥♦s ❞á

q✉❡ u =u′

♦✉ {u, u′

} = {0,1}✳ ❆❣♦r❛✱ s❛❜✐❞♦ q✉❡ cos(2πu) = cos(2πu′

)✱ r❡t♦r♥❛♠♦s

❛ ✐❣✉❛❧❞❛❞❡ ❞❛s ❞✉❛s ♣r✐♠❡✐r❛s ❝♦♦r❞❡♥❛❞❛s ❡ ♦❜t❡♠♦s q✉❡ cos(2πv) = cos(2πv′ ) ❡

sen(2πv) = sen(2πv′

)✱ s❡❣✉✐♥❞♦ q✉❡v =v′

♦✉{v, v′

}={0,1}

P♦rt❛♥t♦✱ (u, v) = (u′, v).

P❛r❛ ♣r♦✈❛r q✉❡g é s♦❜r❡❥❡t♦r❛✱ s❡❥❛ (x, y, z)∈R3 t❛❧ q✉❡ (x2+y22)2+z2 = 1 ❊♥tã♦✱ z ∈ [−1,1]✳ ❈♦♠♦ ❛ ❢✉♥çã♦ s❡♥♦ ❛ss✉♠❡ t♦❞♦s ♦s ✈❛❧♦r❡s ❡♥tr❡ −1 ❡ 1 ♥♦

✐♥t❡r✈❛❧♦ [0,2π]✱ ❡①✐st❡ ϕ [0,2π]t❛❧ q✉❡ z = senϕ✳ ❆ss✐♠✱

x2+y22 = cosϕ

❡✱ ♣♦rt❛♥t♦✱

x2+y2 = 2 + cosϕ.

❋❛③❡♥❞♦ ✉s♦ ❞❡ ❝♦♦r❞❡♥❛❞❛s ♣♦❧❛r❡s✱ ❡①✐st❡θ [0,2π]t❛❧ q✉❡

(x, y) = px2+y2(cosθ,senθ). P♦rt❛♥t♦✱

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✸✹ ❊①❡♠♣❧♦s ❞❡ s✉♣❡r❢í❝✐❡s ❝♦♠♣❛❝t❛s✳

▲♦❣♦✱ t♦♠❛♥❞♦ u= ϕ

2π ∈I ❡v = θ

2π ∈I t❡♠✲s❡ g((u, v)) = (x, y, z)✳

P♦rt❛♥t♦✱ g é s♦❜r❡❥❡t♦r❛✳

▼♦str❡♠♦s ❛❣♦r❛ q✉❡ g é ❝♦♥tí♥✉❛✳

❙❡❥❛ π :I ×I → I×I

∼ ❞❡✜♥✐❞❛ ♣♦r π(u, v) = (u, v) ❛ ♣r♦❥❡çã♦ ♥❛t✉r❛❧✳ ❖ ❡s♣❛ç♦

I×I

∼ ♣♦ss✉✐ ❛ t♦♣♦❧♦❣✐❛ q✉♦❝✐❡♥t❡ ❞❡t❡r♠✐♥❛❞❛ ♣♦r π✳

❱❡❥❛ q✉❡ g◦π :I×I → {(x, y, z)∈R3/(x2+y22)2+z2 = 1} é ❞❛❞❛ ♣♦r

(u, v)7→(p2 + cos(2πu) cos(2πv),p2 + cos(2πu) sen(2πv),sen(2πu)),

q✉❡ é ❝♦♥tí♥✉❛✱ ♣♦✐s ❝❛❞❛ ❢✉♥çã♦ ❝♦♦r❞❡♥❛❞❛ ♦ é✳ ▲♦❣♦✱ ♣❡❧❛ ♣r♦♣r✐❡❞❛❞❡ ✉♥✐✈❡rs❛❧ ❞❡ t♦♣♦❧♦❣✐❛ q✉♦❝✐❡♥t❡✱ ❝♦♥❝❧✉✐✲s❡ q✉❡ g é ❝♦♥tí♥✉❛✳

P❛r❛ ❝♦♥❝❧✉✐r♠♦s q✉❡gé ✉♠ ❤♦♠❡♦♠♦r✜s♠♦✱ ❜❛st❛ ♥♦t❛r♠♦s q✉❡ I×I

∼ é ❝♦♠♣❛❝t♦ ✭♣♦✐s é ✐♠❛❣❡♠ ❞❡ ❝♦♠♣❛❝t♦ ♣♦r ❢✉♥çã♦ ❝♦♥tí♥✉❛✮ ❡ q✉❡

{(x, y, z)∈R3/(x2+y2−2)2+z2 = 1}

é ❞❡ ❍❛✉s❞♦r✛ ✭♣♦✐s é ✉♠ s✉❜❝♦♥❥✉♥t♦ ❞♦ R3✮ ❡ ❛♣❧✐❝❛r ♦ r❡s✉❧t❛❞♦ ❞❡ t♦♣♦❧♦❣✐❛ ❣❡r❛❧

q✉❡ ❞✐③ q✉❡ t♦❞❛ ❜✐❥❡çã♦ ❝♦♥tí♥✉❛ ❞❡ ✉♠ ❡s♣❛ç♦ ❝♦♠♣❛❝t♦ s♦❜r❡ ✉♠ ❡s♣❛ç♦ ❞❡ ❍❛✉s❞♦r✛ é ❤♦♠❡♦♠♦r✜s♠♦✱ ✭❱❡r ❚❡♦r❡♠❛ ✷✮✳

P♦rt❛♥t♦✱ ♣r♦✈❛♠♦s q✉❡ ♦s ❡s♣❛ç♦s {(x, y, z)R3/(x2+y22)2+z2 = 1} I×I ∼ sã♦ ❤♦♠❡♦♠♦r❢♦s✳

❆❣♦r❛✱ ✐r❡♠♦s ♣r♦✈❛r q✉❡ S1×S1 I ×I

∼ sã♦ ❤♦♠❡♦♠♦r❢♦s✳ P❛r❛ t❛❧✱ ❞❡✜♥✐♠♦s ❛ ❢✉♥çã♦ h: I×I

∼ →S

1×S1 ♣♦r

h((u, v)) = (cos(2πu),sen(2πu),cos(2πv),sen(2πv)).

❆ ❢✉♥çã♦ h é ♦❜✈✐❛♠❡♥t❡ ❜❡♠ ❞❡✜♥✐❞❛ ❡ s♦❜r❡❥❡t♦r❛✳

❚❛♠❜é♠✱ ♠♦str❛✲s❡ s❡♠ ❞✐✜❝✉❧❞❛❞❡s q✉❡ h é ✐♥❥❡t♦r❛✳

❆ ❝♦♥t✐♥✉✐❞❛❞❡ ❞❡ h é✱ ❝♦♠♦ ♥♦ ❝❛s♦ ❛♥t❡r✐♦r✱ ✉♠❛ ❛♣❧✐❝❛çã♦ ❞❛ ♣r♦♣r✐❡❞❛❞❡ ✉♥✐✲

✈❡rs❛❧ ❞❡ t♦♣♦❧♦❣✐❛ q✉♦❝✐❡♥t❡✳

P❛r❛ ❝♦♥❝❧✉✐r♠♦s q✉❡ h é ✉♠ ❤♦♠❡♦♠♦r✜s♠♦✱ ♥♦✈❛♠❡♥t❡✱ ❜❛st❛ ♦❜s❡r✈❛r♠♦s q✉❡ I×I

∼ é ❝♦♠♣❛❝t♦ ❡ q✉❡ S

1 ×S1 é ❞❡ ❍❛✉s❞♦r✛✳ P♦rt❛♥t♦✱ ♦s ❡s♣❛ç♦s S1×S1 I×I

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✸✺

❖ ♣ró①✐♠♦ ❡①❡♠♣❧♦ ❞❡ ✉♠❛ s✉♣❡r❢í❝✐❡ ❝♦♠♣❛❝t❛✱ ♠❛s ♥ã♦ ♦r✐❡♥tá✈❡❧✱ q✉❡ ❛♣r❡s❡♥✲ t❛r❡♠♦s é ♦ ♣❧❛♥♦ ♣r♦❥❡t✐✈♦ r❡❛❧✱ ♦✉ s✐♠♣❧❡s♠❡♥t❡ ♣❧❛♥♦ ♣r♦❥❡t✐✈♦✳

❉❡✜♥✐çã♦ ✺✳✶✳ ❖ ❡s♣❛ç♦ q✉♦❝✐❡♥t❡ ❞❛ 2✲❡s❢❡r❛ S2 ♦❜t✐❞♦ ❛tr❛✈és ❞❛ ✐❞❡♥t✐✜❝❛çã♦ ❞❡ ❝❛❞❛ ♣❛r ❞❡ ♣♦♥t♦s ❞✐❛♠❡tr❛❧♠❡♥t❡ ♦♣♦st♦s é ❝❤❛♠❛❞♦ ❞❡ ♣❧❛♥♦ ♣r♦❥❡t✐✈♦✳ ❚♦❞♦ ❡s♣❛ç♦ ❤♦♠❡♦♠♦r❢♦ ❛ ❡st❡ ❡s♣❛ç♦ q✉♦❝✐❡♥t❡ t❛♠❜é♠ é ❝❤❛♠❛❞♦ ❞❡ ♣❧❛♥♦ ♣r♦❥❡t✐✈♦✳

❙❡❥❛H ={(x, y, z)S2 |z 0} ♦ ❤❡♠✐s❢ér✐♦ s✉♣❡r✐♦r ❢❡❝❤❛❞♦ ❞❡ S2✳ ➱ ❝❧❛r♦ q✉❡✱ ❞❡ ❝❛❞❛ ♣❛r ❞❡ ♣♦♥t♦s ❞✐❛♠❡tr❛❧♠❡♥t❡ ♦♣♦st♦s ❡♠ S2✱ ♣❡❧♦ ♠❡♥♦s ✉♠ ♣♦♥t♦ ❡stá ❡♠

H✳ ❙❡ ♦s ❞♦✐s ♣♦♥t♦s ❡stã♦ ❡♠ H✱ ❡♥tã♦ ❡❧❡s ❡stã♦ ♥♦ ❡q✉❛❞♦r✱ q✉❡ é ♦ ❜♦r❞♦ ❞❡ H✳

❆ss✐♠✱ ♣♦❞❡rí❛♠♦s t❛♠❜é♠ ❞❡✜♥✐r ♦ ♣❧❛♥♦ ♣r♦❥❡t✐✈♦ ❝♦♠♦ ♦ ❡s♣❛ç♦ q✉♦❝✐❡♥t❡ ❞❡ H

♦❜t✐❞♦ ♣❡❧❛ ✐❞❡♥t✐✜❝❛çã♦ ❞♦s ♣♦♥t♦s ❞✐❛♠❡tr❛❧♠❡♥t❡ ♦♣♦st♦s q✉❡ ❡stã♦ ♥♦ ❜♦r❞♦ ❞❡ H✳

❈♦♠♦ H é ❤♦♠❡♦♠♦r❢♦ ❛♦ ❞✐s❝♦ ✉♥✐tár✐♦ ❢❡❝❤❛❞♦E2 ❞♦ ♣❧❛♥♦✱

E2 ={(x, y)R2 | x2+y2 1},

♦ ❡s♣❛ç♦ q✉♦❝✐❡♥t❡ ❞❡ E2 ♦❜t✐❞♦ ♣❡❧❛ ✐❞❡♥t✐✜❝❛çã♦ ❞♦s ♣♦♥t♦s ❞❡ S1 q✉❡ sã♦ ❞✐❛♠❡✲ tr❛❧♠❡♥t❡ ♦♣♦st♦s é ✉♠ ♣❧❛♥♦ ♣r♦❥❡t✐✈♦✳ P♦r E2✱ ♣♦❞❡♠♦s s✉❜st✐t✉✐r q✉❛❧q✉❡r ❡s♣❛ç♦ ❤♦♠❡♦♠♦r❢♦✱ ♣♦r ❡①❡♠♣❧♦✱ ✉♠ q✉❛❞r❛❞♦✳

❙❡❥❛X ♦ q✉❛❞r❛❞♦ ✉♥✐tár✐♦ ♥♦ ♣❧❛♥♦ R2

{(x, y)R2 | 0x1,0y1}.

❖ ♣❧❛♥♦ ♣r♦❥❡t✐✈♦ é q✉❛❧q✉❡r ❡s♣❛ç♦ ❤♦♠❡♦♠♦r❢♦ ❛♦ ❡s♣❛ç♦ q✉♦❝✐❡♥t❡ ❞❡X ♦❜t✐❞♦ ♣❡❧❛

✐❞❡♥t✐✜❝❛çã♦ ❞♦s ❧❛❞♦s ♦♣♦st♦s ❞♦ q✉❛❞r❛❞♦X ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛s s❡❣✉✐♥t❡s r❡❣r❛s✿

✶✳ P❛r❛ ❝❛❞❛ y ♥♦ ✐♥t❡r✈❛❧♦[0,1]✱ ♦s ♣♦♥t♦s (0, y) ❡ (1,1y) s❡ ✐❞❡♥t✐✜❝❛♠✳

✷✳ P❛r❛ ❝❛❞❛ x ♥♦ ✐♥t❡r✈❛❧♦[0,1]✱ ♦s ♣♦♥t♦s (x,0)❡ (1x,1) s❡ ✐❞❡♥t✐✜❝❛♠✳

❊ss❛s ✐❞❡♥t✐✜❝❛çõ❡s ❡stã♦ r❡♣r❡s❡♥t❛❞❛s ♥♦ ❞✐❛❣r❛♠❛ ❛❜❛✐①♦✿

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✸✻ ❊①❡♠♣❧♦s ❞❡ s✉♣❡r❢í❝✐❡s ❝♦♠♣❛❝t❛s✳

P♦❞❡♠♦s ✈❡r q✉❡ ♦ ♣❧❛♥♦ ♣r♦❥❡t✐✈♦ é ♥ã♦✲♦r✐❡♥tá✈❡❧ ♣❡❧❛ ♦❜s❡r✈❛çã♦ ❢❡✐t❛ ♥❛ ✜❣✉r❛ ✺✳✸✱ ❞❡ q✉❡ ❡❧❡ ❝♦♥té♠ ✉♠❛ ❢❛✐①❛ ❞❡ ▼ö❜✐✉s✳

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✻ ❙♦♠❛ ❝♦♥❡①❛

❉❡s❝r❡✈❡r❡♠♦s ❝♦♠♦ ❞❛r ♦✉tr♦s ❡①❡♠♣❧♦s ❞❡ s✉♣❡r❢í❝✐❡s ❝♦♠♣❛❝t❛s ❛♦ ❢♦r♠❛r ♦ q✉❡ sã♦ ❝❤❛♠❛❞❛s s♦♠❛s ❝♦♥❡①❛s✳

❙❡❥❛♠ S1 ❡ S2 s✉♣❡r❢í❝✐❡s ❞✐s❥✉♥t❛s✳ ❙✉❛ s♦♠❛ ❝♦♥❡①❛✱ ❞❡♥♦t❛❞❛ ♣♦r S1#S2✱ é ❢♦r♠❛❞❛ ❝♦rt❛♥❞♦ ❡ r❡♠♦✈❡♥❞♦ ✉♠❛ ♣❡q✉❡♥❛ r❡❣✐ã♦ ❝✐r❝✉❧❛r ❞❡ ❝❛❞❛ ✉♠❛ ❞❛s ❞✉❛s s✉♣❡r❢í❝✐❡s✳ ■st♦ ❝r✐❛rá ✉♠ ♣❡q✉❡♥♦ ❜♦r❞♦ ❝✐r❝✉❧❛r ❡♠ ❝❛❞❛ ✉♠❛ ❞❡❧❛s ❡ ❡♥tã♦ ❝♦❧❛✲s❡ ❛s ❞✉❛s s✉♣❡r❢í❝✐❡s ❛♦ ❧♦♥❣♦ ❞♦s ❜♦r❞♦s ❝✐r❝✉❧❛r❡s✳

P❛r❛ s❡r ♠❛✐s ♣r❡❝✐s♦✱ ❡s❝♦❧❤❡♠♦s s✉❜❝♦♥❥✉♥t♦s D1 ⊂ S1 ❡ D2 ⊂ S2 t❛❧ q✉❡ D1 ❡

D2 sã♦ ❞✐s❝♦s ❢❡❝❤❛❞♦s ✭✐st♦ é✱ ❤♦♠❡♦♠♦r❢♦ ❛ E2✮✳ ❙❡❥❛ Si′ ♦ ❝♦♠♣❧❡♠❡♥t♦ ❞♦ ✐♥t❡r✐♦r ❞❡ Di ❡♠ Si ♣❛r❛ i= 1 ❡ 2✳

❊s❝♦❧❤❛ ✉♠ ❤♦♠❡♦♠♦r✜s♠♦h ❞❛ ❢r♦♥t❡✐r❛ ❞❡ D1 s♦❜r❡ ❛ ❢r♦♥t❡✐r❛ ❞❡ D2✳

❊♥tã♦S1#S2 é ♦ ❡s♣❛ç♦ q✉♦❝✐❡♥t❡ ❞❡S1′ ∪S2′ ♦❜t✐❞♦ ♣❡❧❛ ✐❞❡♥t✐✜❝❛çã♦ ❞♦s ♣♦♥t♦s

x ❡ h(x)✱ ♣❛r❛ t♦❞♦s ♦s ♣♦♥t♦s x ❞❛ ❢r♦♥t❡✐r❛ ❞❡ D1✱ ❝♦♠S1#S2 r❡s✉❧t❛♥❞♦ ✉♠❛ ♥♦✈❛ s✉♣❡r❢í❝✐❡✳

❊①❡♠♣❧♦ ✻✳✶✳ ❙♦♠❛ ❈♦♥❡①❛ ❞❡ ❚♦r♦s✳

❙❡❥❛♠T1 ❡T2 t♦r♦s✱ r❡♣r❡s❡♥t❛❞♦s ❝❛❞❛ ✉♠ ♣♦r ✉♠ q✉❛❞r❛❞♦ ❝♦♠ ♦s ❧❛❞♦s ♦♣♦st♦s ✐❞❡♥t✐✜❝❛❞♦s ❝♦♠♦ ♠♦str❛ ❛ ✜❣✉r❛ ✻✳✶✭❛✮✱ s❡♥❞♦ ♦s q✉❛tr♦ ✈ért✐❝❡s ❞❡ ❝❛❞❛ q✉❛❞r❛❞♦ ✐❞❡♥t✐✜❝❛❞♦s ❡♠ ✉♠ só ♣♦♥t♦ ♥♦ t♦r♦ ❝♦rr❡s♣♦♥❞❡♥t❡✳

P❛r❛ ❢♦r♠❛r ❛ s✉❛ s♦♠❛ ❝♦♥❡①❛✱ ❞❡✈❡♠♦s ♣r✐♠❡✐r❛♠❡♥t❡ r❡❝♦rt❛r ✉♠ ❜✉r❛❝♦ ❝✐r❝✉❧❛r ❡♠ ❝❛❞❛ t♦r♦✱ ♥♦♠❡❛♥❞♦ ❞❡c1 ❡c2 ❛s ❢r♦♥t❡✐r❛s ❞❡st❡s ❜✉r❛❝♦s✱ ❡ ❡❧❡s sã♦ ✐❞❡♥t✐✜❝❛❞♦s ❝♦♠♦ ✐♥❞✐❝❛❞♦ ♣❡❧❛s s❡t❛s✳ P♦❞❡♠♦s t❛♠❜é♠ r❡♣r❡s❡♥t❛r ♦ ❝♦♠♣❧❡♠❡♥t♦ ❞♦s ❜✉r❛❝♦s ♥♦s ❞♦✐s t♦r♦s ♣♦r ♣❡♥tá❣♦♥♦s ♠♦str❛❞♦s ♥❛ ✜❣✉r❛ ✻✳✶✭❜✮✳

■❞❡♥t✐✜❝❛♠♦s ♦s s❡❣♠❡♥t♦sc1 ❡c2✱ ♦❜t❡♥❞♦ ♦ ♦❝tó❣♦♥♦ ❞❛ ✜❣✉r❛ ✻✳✶✭❝✮✱ ♥♦ q✉❛❧ ♦s ❧❛❞♦s ❡stã♦ ✐❞❡♥t✐✜❝❛❞♦s ❛♦s ♣❛r❡s✱ s❡❣✉♥❞♦ ❛s ♦r✐❡♥t❛çõ❡s ❞❛s ❛r❡st❛s✳

◆♦t❡ q✉❡ t♦❞♦s ♦s ♦✐t♦ ✈ért✐❝❡s ❞❡st❡ ♦❝tó❣♦♥♦ sã♦ ✐❞❡♥t✐✜❝❛❞♦s ♥✉♠ ú♥✐❝♦ ♣♦♥t♦ ❡♠ T1#T2✳

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✸✽ ❙♦♠❛ ❝♦♥❡①❛

❋✐❣✉r❛ ✻✳✶✿ ❙♦♠❛ ❈♦♥❡①❛ ❞❡ ❞♦✐s t♦r♦s✳

❘❡♣❡t✐♥❞♦ ❡ss❡ ♣r♦❝❡ss♦✱ ♣♦❞❡♠♦s ♠♦str❛r q✉❡ ❛ s♦♠❛ ❝♦♥❡①❛ ❞❡ três t♦r♦s é ♦ ❡s♣❛ç♦ q✉♦❝✐❡♥t❡ ❞♦ ❞♦❞❡❝á❣♦♥♦✱ q✉❛♥❞♦ ❛s ❛r❡st❛s sã♦ ✐❞❡♥t✐✜❝❛❞❛s ❛♦s ♣❛r❡s ❝♦♠♦ ✐♥❞✐❝❛❞♦ ♥❛ ✜❣✉r❛ ✻✳✷

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✸✾

P♦r ✐♥❞✉çã♦ ✜♥✐t❛✱ ♣r♦✈❛r❡♠♦s q✉❡✱ ♣❛r❛ q✉❛❧q✉❡r ✐♥t❡✐r♦ ♣♦s✐t✐✈♦n✱ ❛ s♦♠❛ ❝♦♥❡①❛

❞❡n✲t♦r♦s é ❤♦♠❡♦♠♦r❢❛ ❛♦ ❡s♣❛ç♦ q✉♦❝✐❡♥t❡ ❞❡ ✉♠ ♣♦❧í❣♦♥♦ ❞❡4n❧❛❞♦s✱ ❝✉❥❛s ❛r❡st❛s

❡stã♦ ✐❞❡♥t✐✜❝❛❞❛s ❛♦s ♣❛r❡s✱ ♦❜❡❞❡❝❡♥❞♦ ❝❡rt❛ ♦r✐❡♥t❛çã♦ ❞❛s ❛r❡st❛s ♥♦s ♣♦❧í❣♦♥♦s✳ ❈♦♥s✐❞❡r❡♠♦sS =T2#...#T2 ❛ s♦♠❛ ❝♦♥❡①❛ ❞❡ n t♦r♦s✱ n 2

❙❡n = 2✱ s❛❜❡♠♦s ♣❡❧❛ ✜❣✉r❛ ✻✳✶✱ q✉❡ ❛ s♦♠❛ ❝♦♥❡①❛ ❞❡ ❞♦✐s t♦r♦s é ❤♦♠❡♦♠♦r❢❛

❛♦ ❡s♣❛ç♦ q✉♦❝✐❡♥t❡ ❞♦ ♦❝tó❣♦♥♦✱ ♦✉ s❡❥❛✱ é ❤♦♠❡♦♠♦r❢❛ ❛♦ ❡s♣❛ç♦ q✉♦❝✐❡♥t❡ ❞❡ ✉♠ ♣♦❧í❣♦♥♦ ❞❡ 4×2 ❧❛❞♦s✳

❙✉♣♦♥❤❛♠♦s q✉❡ ❛ ❛✜r♠❛çã♦ s❡❥❛ ✈á❧✐❞❛ ♣❛r❛ ✉♠ ❝❡rt♦ n✳ ❆ s♦♠❛ ❝♦♥❡①❛ ❞❡ n

t♦r♦s é ❤♦♠❡♦♠♦r❢❛ ❛♦ ❡s♣❛ç♦ q✉♦❝✐❡♥t❡ ❞❡ ✉♠ ♣♦❧í❣♦♥♦ ❞❡ 4n ❧❛❞♦s✳

❙❡❥❛ S = T2#...#T2 ❛ s♦♠❛ ❝♦♥❡①❛ ❞❡ (n + 1) t♦r♦s✱ q✉❡ ♣♦❞❡ s❡r ✈✐st❛ ❝♦♠♦

S = (T2#...#T2)#T2✱ ❛ s♦♠❛ ❝♦♥❡①❛ ❞❡ n✲t♦r♦s ❝♦♠ ✉♠ t♦r♦✳

P❡❧❛ ❤✐♣ót❡s❡ ❞❡ ✐♥❞✉çã♦✱ t❡♠♦s q✉❡ ❛ s♦♠❛ ❝♦♥❡①❛ ❞❡ n✲t♦r♦s é ❤♦♠❡♦♠♦r❢❛ ❛♦

❡s♣❛ç♦ q✉♦❝✐❡♥t❡ ❞❡ ✉♠ ♣♦❧í❣♦♥♦ ❞❡ 4n ❧❛❞♦s✳ ❋❛③❡♥❞♦ ❛ s♦♠❛ ❝♦♥❡①❛ ❞♦ ♣♦❧í❣♦♥♦ ❞❡

4n ❧❛❞♦s ❝♦♠ ✉♠ t♦r♦ ♦❜t❡♠♦s ✉♠ ♣♦❧í❣♦♥♦ ❞❡ (4n+ 4) ❧❛❞♦s✱ ♦✉ s❡❥❛✱ ❛ s♦♠❛ ❝♦♥❡①❛

❞❡ (n+ 1) t♦r♦s é ❤♦♠❡♦♠♦r❢❛ ❛♦ ❡s♣❛ç♦ q✉♦❝✐❡♥t❡ ❞❡ ✉♠ ♣♦❧í❣♦♥♦ ❞❡4(n+ 1) ❧❛❞♦s✳

P♦rt❛♥t♦✱ ♣♦r ✐♥❞✉çã♦ ✜♥✐t❛✱ ❛ s♦♠❛ ❝♦♥❡①❛ ❞❡ n t♦r♦s é ❤♦♠❡♦♠♦r❢❛ ❛♦ ❡s♣❛ç♦

q✉♦❝✐❡♥t❡ ❞❡ ✉♠ ♣♦❧í❣♦♥♦ ❞❡ 4n ❧❛❞♦s✳

❊①❡♠♣❧♦ ✻✳✷✳ ❙♦♠❛ ❈♦♥❡①❛ ❞❡ P❧❛♥♦s Pr♦❥❡t✐✈♦s✳

❖ ♣❧❛♥♦ ♣r♦❥❡t✐✈♦ é ❞❡✜♥✐❞♦ ❝♦♠♦ s❡♥❞♦ ♦ ❡s♣❛ç♦ q✉♦❝✐❡♥t❡ ❞❡ ✉♠ ❞✐s❝♦ ❝✐r❝✉❧❛r✱ ✐❞❡♥t✐✜❝❛♥❞♦ ♦s ♣♦♥t♦s ❞✐❛♠❡tr❛❧♠❡♥t❡ ♦♣♦st♦s ♥❛ ❢r♦♥t❡✐r❛✳ ❆♦ ❡s❝♦❧❤❡r ✉♠ ♣❛r ❞❡ ♣♦♥t♦s ❞✐❛♠❡tr❛❧♠❡♥t❡ ♦♣♦st♦s ♥❛ ❢r♦♥t❡✐r❛ ❝♦♠♦ ✈ért✐❝❡s✱ ❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞♦ ❞✐s❝♦ ❝✐r❝✉❧❛r é ❞✐✈✐❞✐❞❛ ❡♠ ❞♦✐s s❡❣♠❡♥t♦s✳ ❆ss✐♠✱ ♣♦❞❡♠♦s ❝♦♥s✐❞❡r❛r ♦ ♣❧❛♥♦ ♣r♦❥❡t✐✈♦ ❝♦♠♦ ♦❜t✐❞♦ ❞❡ ✉♠ ♣♦❧í❣♦♥♦ ♣❡❧❛ ✐❞❡♥t✐✜❝❛çã♦ ❞❡ s❡✉s ❧❛❞♦s ♦♣♦st♦s ❝♦♠♦ ♠♦str❛ ❛ ✜❣✉r❛ ✻✳✸

❋✐❣✉r❛ ✻✳✸✿ ❈♦♥str✉çã♦ ❞♦ ♣❧❛♥♦ ♣r♦❥❡t✐✈♦ ❝♦♠♦ ❡s♣❛ç♦ q✉♦❝✐❡♥t❡ ❞❡ ✉♠ ♣♦❧í❣♦♥♦ ❞❡ ❞♦✐s ❧❛❞♦s✳

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✹✵ ❙♦♠❛ ❝♦♥❡①❛

♦ ♠❡s♠♦ ✉s❛❞♦ ♣❛r❛ ♦❜t❡r ❛ r❡s♣r❡s❡♥t❛çã♦ ❞❛ s♦♠❛ ❝♦♥❡①❛ ❞❡ ❞♦✐s t♦r♦s ❝♦♠♦ ✉♠ ❡s♣❛ç♦ q✉♦❝✐❡♥t❡ ❞❡ ✉♠ ♦❝tó❣♦♥♦✳

❋✐❣✉r❛ ✻✳✹✿ ❙♦♠❛ ❈♦♥❡①❛ ❞❡ ❞♦✐s ♣❧❛♥♦s ♣r♦❥❡t✐✈♦s✳

❆♦ r❡♣❡t✐r ❡st❡ ♣r♦❝❡ss♦✱ ✈❡♠♦s q✉❡ ❛ s♦♠❛ ❝♦♥❡①❛ ❞❡ três ♣❧❛♥♦s ♣r♦❥❡t✐✈♦s é ♦ ❡s♣❛ç♦ q✉♦❝✐❡♥t❡ ❞❡ ✉♠ ❤❡①á❣♦♥♦ ❝♦♠ ♦s ❧❛❞♦s ✐❞❡♥t✐✜❝❛❞♦s ❛♦s ♣❛r❡s✱ ❝♦♠♦ ♠♦str❛ ❛ ✜❣✉r❛ ✻✳✺

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✹✶

P♦r ✐♥❞✉çã♦ ✜♥✐t❛ ✱ ♣r♦✈❛r❡♠♦s q✉❡✱ ♣❛r❛ q✉❛❧q✉❡r ✐♥t❡✐r♦ ♣♦s✐t✐✈♦n✱ ❛ s♦♠❛ ❝♦♥❡①❛

❞❡ n✲♣❧❛♥♦s ♣r♦❥❡t✐✈♦s é ♦ ❡s♣❛ç♦ q✉♦❝✐❡♥t❡ ❞❡ ✉♠ ♣♦❧í❣♦♥♦ ❞❡ 2n ❧❛❞♦s✱ ❝♦♠ ♦s ❧❛❞♦s

✐❞❡♥t✐✜❝❛❞♦s ❛♦s ♣❛r❡s✱ ♦❜❡❞❡❝✐❞❛ ❛ ♦r✐❡♥t❛çã♦ ❞❛s ❛r❡st❛s ❞♦ ♣♦❧í❣♦♥♦✳ ❖♥❞❡ t♦❞♦s ♦s ✈ért✐❝❡s ❞❡st❡ ♣♦❧í❣♦♥♦ sã♦ ✐❞❡♥t✐✜❝❛❞♦s ❡♠ ✉♠ ú♥✐❝♦ ♣♦♥t♦✳

❈♦♥s✐❞❡r❡♠♦sS =P2#...#P2 ❛ s♦♠❛ ❝♦♥❡①❛ ❞❡ n ♣❧❛♥♦s ♣r♦❥❡t✐✈♦s✱ n 2 ❙❡ n = 2✱ s❛❜❡♠♦s ♣❡❧❛ ✜❣✉r❛ ✻✳✹ q✉❡ ❛ s♦♠❛ ❝♦♥❡①❛ ❞❡ ❞♦✐s ♣❧❛♥♦s ♣r♦❥❡t✐✈♦s

é ❤♦♠❡♦♠♦r❢❛ ❛♦ ❡s♣❛ç♦ q✉♦❝✐❡♥t❡ ❞♦ q✉❛❞r✐❧át❡r♦✱ ♦✉ s❡❥❛✱ é ❤♦♠❡♦♠♦r❢❛ ❛♦ ❡s♣❛ç♦ q✉♦❝✐❡♥t❡ ❞❡ ✉♠ ♣♦❧í❣♦♥♦ ❞❡ 2×2 ❧❛❞♦s✳

❙✉♣♦♥❤❛♠♦s q✉❡ ❛ ❛✜r♠❛çã♦ s❡❥❛ ✈á❧✐❞❛ ♣❛r❛ ✉♠ ❝❡rt♦ n✳ ❆ s♦♠❛ ❝♦♥❡①❛ ❞❡ n

♣❧❛♥♦s ♣r♦❥❡t✐✈♦s é ❤♦♠❡♦♠♦r❢❛ ❛♦ ❡s♣❛ç♦ q✉♦❝✐❡♥t❡ ❞❡ ✉♠ ♣♦❧í❣♦♥♦ ❞❡2n ❧❛❞♦s✳

❙❡❥❛S =P2#...#P2 ❛ s♦♠❛ ❝♦♥❡①❛ ❞❡(n+ 1) ♣❧❛♥♦s ♣r♦❥❡t✐✈♦s✱ q✉❡ ♣♦❞❡ s❡r ✈✐st❛ ❝♦♠♦ S = (P2#...#P2)#P2✱ ❛ s♦♠❛ ❝♦♥❡①❛ ❞❡ n ♣❧❛♥♦s ♣r♦❥❡t✐✈♦s ❝♦♠ ✉♠ ♣❧❛♥♦ ♣r♦❥❡t✐✈♦✳

P❡❧❛ ❤✐♣ót❡s❡ ❞❡ ✐♥❞✉çã♦✱ t❡♠♦s q✉❡ ❛ s♦♠❛ ❝♦♥❡①❛ ❞❡n ♣❧❛♥♦s ♣r♦❥❡t✐✈♦s é ❤♦♠❡✲

♦♠♦r❢❛ ❛♦ ❡s♣❛ç♦ q✉♦❝✐❡♥t❡ ❞❡ ✉♠ ♣♦❧í❣♦♥♦ ❞❡ 2n ❧❛❞♦s✳ ❋❛③❡♥❞♦ ❛ s♦♠❛ ❝♦♥❡①❛ ❞♦

♣♦❧í❣♦♥♦ ❞❡ 2n ❧❛❞♦s ❝♦♠ ✉♠ ♣❧❛♥♦ ♣r♦❥❡t✐✈♦ ♦❜t❡♠♦s ✉♠ ♣♦❧í❣♦♥♦ ❞❡ (2n+ 2)❧❛❞♦s✱

♦✉ s❡❥❛✱ ❛ s♦♠❛ ❝♦♥❡①❛ ❞❡ (n+ 1) ♣❧❛♥♦s ♣r♦❥❡t✐✈♦s é ❤♦♠❡♦♠♦r❢❛ ❛♦ ❡s♣❛ç♦ q✉♦❝✐❡♥t❡

❞❡ ✉♠ ♣♦❧í❣♦♥♦ ❞❡ 2(n+ 1) ❧❛❞♦s✳

P♦rt❛♥t♦✱ ♣♦r ✐♥❞✉çã♦ ✜♥✐t❛✱ ❛ s♦♠❛ ❝♦♥❡①❛ ❞❡n ♣❧❛♥♦s ♣r♦❥❡t✐✈♦s é ❤♦♠❡♦♠♦r❢❛

❛♦ ❡s♣❛ç♦ q✉♦❝✐❡♥t❡ ❞❡ ✉♠ ♣♦❧í❣♦♥♦ ❞❡ 2n ❧❛❞♦s✳

❊①❡♠♣❧♦ ✻✳✸✳ ❙♦♠❛ ❈♦♥❡①❛ ❞❡ ✉♠❛ ❙✉♣❡r❢í❝✐❡ ❝♦♠ ✉♠❛ ❊s❢❡r❛✳

❙❡❥❛♠ S2 ❛ ❡s❢❡r❛ ❜✐❞✐♠❡♥s✐♦♥❛❧ ❡ M ✉♠❛ s✉♣❡r❢í❝✐❡ q✉❛❧q✉❡r✳ ❊♥t❛♦ M#S2 é ❤♦♠❡♦♠♦r❢❛ ❛ M✳

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✹✷ ❙♦♠❛ ❝♦♥❡①❛

❘❡♣r❡s❡♥t❛r❡♠♦s ❛ ❡s❢❡r❛ ❝♦♠♦ ♦ ❡s♣❛ç♦ q✉♦❝✐❡♥t❡ ❞❡ ✉♠ ♣♦❧í❣♦♥♦ ❝♦♠ ❞♦✐s ❧❛❞♦s ✐❞❡♥t✐✜❝❛❞♦s ❛♦s ♣❛r❡s ❝♦♠♦ ♠♦str❛ ❛ ✜❣✉r❛ ✻✳✼✳

❋✐❣✉r❛ ✻✳✼✿ ❈♦♥str✉çã♦ ❞❛ ❡s❢❡r❛✳

P♦❞❡♠♦s ✐♠❛❣✐♥❛r ✉♠❛ ❡s❢❡r❛ ❝♦♠ ✉♠ ③í♣❡r s♦❜r❡ ❡❧❛✱ ❝♦♠♦ ✉♠❛ ❜♦❧s❛✱ q✉❛♥❞♦ ♦ ③í♣❡r é ❛❜❡rt♦ ❛ ❜♦❧s❛ ♣♦❞❡ t♦r♥❛r✲s❡ ♣❧❛♥❛✳

❊①❡♠♣❧♦ ✻✳✹✳ ❙❡ S1 ❡ S2 sã♦ ♣❧❛♥♦s ♣r♦❥❡t✐✈♦s ❡♥tã♦ S1#S2 é ✉♠❛ ✧●❛rr❛❢❛ ❞❡ ❑❧❡✐♥✧✱ ✐st♦ é✱ ❤♦♠❡♦♠♦r❢❛ ❛ s✉♣❡r❢í❝✐❡ ♦❜t✐❞❛ ♣❡❧❛ ✐❞❡♥t✐✜❝❛çã♦ ❞♦s ❧❛❞♦s ♦♣♦st♦s ❞❡ ✉♠ q✉❛❞r❛❞♦ ❝♦♠♦ ♥❛ ✜❣✉r❛ ✻✳✽✳

❋✐❣✉r❛ ✻✳✽✿ ●❛rr❛❢❛ ❞❡ ❑❧❡✐♥✳

✶✳ P❛r❛ ❝❛❞❛ y ♥♦ ✐♥t❡r✈❛❧♦[0,1]✱ ♦s ♣♦♥t♦s (0, y) ❡(1,1−y)s❡ ✐❞❡♥t✐✜❝❛♠✳

✷✳ P❛r❛ ❝❛❞❛ x♥♦ ✐♥t❡r✈❛❧♦ [0,1]✱ ♦s ♣♦♥t♦s (x,0)❡ (x,1)s❡ ✐❞❡♥t✐✜❝❛♠✳

P♦❞❡♠♦s ♣r♦✈❛r ✐st♦ ♣❡❧❛ té❝♥✐❝❛ ❞♦ ✧❝♦rt❛r✧ ❡ ✧❝♦❧❛r✧✱ ❝♦♠♦ s❡❣✉❡✳ ❙❡Si é ♦ ♣❧❛♥♦ ♣r♦❥❡t✐✈♦✱ ❡Di é ✉♠ ❞✐s❝♦ ❢❡❝❤❛❞♦ t❛❧ q✉❡Di ⊂Si✱ ❡♥tã♦Si′✱ ♦ ❝♦♠♣❧❡♠❡♥t♦ ❞♦ ✐♥t❡r✐♦r ❞❡Di ❡♠ Si✱ é ❤♦♠❡♦♠♦r❢♦ ❛ ✉♠❛ ❢❛✐①❛ ❞❡ ▼ö❜✐✉s ✭✐♥❝❧✉✐♥❞♦ ♦ ❜♦r❞♦✮✱ i= 1,2✳

❉❡ ❢❛t♦✱ s❡ ♣❡♥s❛r♠♦s Si ❝♦♠♦ ♦ ❡s♣❛ç♦ ♦❜t✐❞♦ ♣❡❧❛ ✐❞❡♥t✐✜❝❛çã♦ ❞♦s ♣♦♥t♦s ❞✐❛✲ ♠❡tr❛❧♠❡♥t❡ ♦♣♦st♦s ♥❛ ❜♦r❞❛ ❞♦ ❞✐s❝♦ ✉♥✐tár✐♦E2 ❡♠ R2✱ ❡♥tã♦ ♣♦❞❡♠♦s ❡s❝♦❧❤❡rD

i ♣❛r❛ s❡r ❛ ✐♠❛❣❡♠ ❞♦ ❝♦♥❥✉♥t♦ {(x, y)∈E2 :|y |≥ 1

Referências

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