A característica de Euler

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

❈❛♠♣✉s ❞❡ ❘✐♦ ❈❧❛r♦

❆ ❈❛r❛❝t❡ríst✐❝❛ ❞❡ ❊✉❧❡r

❉❡♥✐s ❱❛♥✉❝❝✐ ●✐s♦❧❞✐

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

❖r✐❡♥t❛❞♦r❛

Pr♦❢❛✳ ❉r❛✳ ❆❧✐❝❡ ❑✐♠✐❡ ▼✐✇❛ ▲✐❜❛r❞✐

✺✶✹✳✷ ●✺✸✺❝

●✐s♦❧❞✐✱ ❉❡♥✐s ❱❛♥✉❝❝✐

❆ ❈❛r❛❝t❡ríst✐❝❛ ❞❡ ❊✉❧❡r✴ ❉❡♥✐s ❱❛♥✉❝❝✐ ●✐s♦❧❞✐✲ ❘✐♦ ❈❧❛r♦✱ ✷✵✶✸✳

✹✾ ❢✳✿ ✐❧✳✱ ✜❣s✳✱ ❣rá❢s✳

❉✐ss❡rt❛çã♦ ✭♠❡str❛❞♦✮ ✲ ❯♥✐✈❡rs✐❞❛❞❡ ❊st❛❞✉❛❧ P❛✉❧✐st❛✱ ■♥st✐✲ t✉t♦ ❞❡ ●❡♦❝✐ê♥❝✐❛s ❡ ❈✐ê♥❝✐❛s ❊①❛t❛s✳

❖r✐❡♥t❛❞♦r❛✿ ❆❧✐❝❡ ❑✐♠✐❡ ▼✐✇❛ ▲✐❜❛r❞✐

✶✳ ❚♦♣♦❧♦❣✐❛ ❛❧❣é❜r✐❝❛✳ ✷✳ P♦❧✐❡❞r♦s✳ ✸✳ ❙✉♣❡r❢í❝✐❡s✳ ✹✳ ❈❧❛ss✐✜✲ ❝❛çã♦ ❞❡ ❙✉♣❡r❢í❝✐❡s✳ ■✳ ❚ít✉❧♦

❚❊❘▼❖ ❉❊ ❆P❘❖❱❆➬➹❖

❉❡♥✐s ❱❛♥✉❝❝✐ ●✐s♦❧❞✐

❆ ❈❛r❛❝t❡ríst✐❝❛ ❞❡ ❊✉❧❡r

❉✐ss❡rt❛çã♦ ❛♣r♦✈❛❞❛ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ▼❡str❡ ♥♦ ❈✉rs♦ ❞❡ Pós✲●r❛❞✉❛çã♦ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛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❛✳ ❆❧✐❝❡ ❑✐♠✐❡ ▼✐✇❛ ▲✐❜❛r❞✐ ❖r✐❡♥t❛❞♦r❛

Pr♦❢✳ ❉r✳ ▼❛r❝♦s ❱✐❡✐r❛ ❚❡✐①❡✐r❛ ■●❈❊✴❯◆❊❙P✴❘✐♦ ❈❧❛r♦✭❙P✮

Pr♦❢❛✳ ❉r❛✳ ❊✈❡❧✐♥ ▼❡♥❡❣✉❡ss♦ ❇❛r❜❛r❡s❝♦ ■❇■▲❈❊✴❯◆❊❙P✴❙ã♦ ❏♦sé ❞♦ ❘✐♦ Pr❡t♦✭❙P✮

❆❣r❛❞❡❝✐♠❡♥t♦s

❆❣r❛❞❡ç♦ ❛♦ ●♦✈❡r♥♦ ❋❡❞❡r❛❧✱ ❛♦ ▼❊❈ ❡ à ❙♦❝✐❡❞❛❞❡ ❇r❛s✐❧❡✐r❛ ❞❡ ▼❛t❡♠át✐❝❛ ♣❡❧❛ ✐♥✐❝✐❛t✐✈❛ ♣✐♦♥❡✐r❛ ❞❡ r❡❛❧✐③❛çã♦ ❞❡ss❡ ♣r♦❥❡t♦ ❡♠ r❡❞❡ ♥❛❝✐♦♥❛❧✱ ❛ ❈❆P❊❙ ♣❡❧♦ ✜♥❛♥❝✐❛♠❡♥t♦ ❞❡ ❜♦❧s❛s ❞❡ ❡st✉❞♦s✱ ♣♦✐s ♥ã♦ s❡r✐❛ ♣♦ssí✈❡❧ s✉❛ ❡❢❡t✐✈❛çã♦ s❡♠ ❡st❡ ❛♣♦✐♦ ❡ ❛ t♦❞♦s ♦s ❡♥✈♦❧✈✐❞♦s ♥❛ ❯♥✐✈❡rs✐❞❛❞❡ ❊st❛❞✉❛❧ P❛✉❧✐st❛ ✧❏✉❧✐♦ ❞❡ ▼❡sq✉✐t❛ ❋✐❧❤♦✧ ♣♦r ♥ã♦ t❡r❡♠ ♠❡❞✐❞♦ ❡s❢♦rç♦s ♣❛r❛ ❢❛③❡r ♣❛rt❡ ❞❡st❡ ♣r♦❥❡t♦✳ ❆❣r❛❞❡ç♦ ❛ t♦❞❛s ❛s ♣❡ss♦❛s q✉❡✱ ❞✐r❡t❛ ♦✉ ✐♥❞✐r❡t❛♠❡♥t❡✱ ❝♦♥tr✐❜✉ír❛♠ ♣❛r❛ ❛ r❡❛❧✐③❛çã♦ ❞❡st❡ tr❛❜❛❧❤♦✱ ❡♠ ♣❛rt✐❝✉❧❛r✱ à ❝♦♦r❞❡♥❛❞♦r❛ ❧♦❝❛❧✱ ❙✉③✐♥❡✐ ▼❛r❝♦♥❛t♦✱ ♣♦r s❡♠♣r❡ ♠❡ ♠♦t✐✈❛r ♥♦s ♠♦♠❡♥t♦s ♠❛✐s ❞✐❢í❝❡✐s ♥♦ ❝✉rs♦✳

❆♦s ♠❡✉s ❛♠✐❣♦s ❞❡ ❝✉rs♦✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡✱ ▲❡❛♥❞r♦ ❚❡③♦tt♦ ❡ ▼❛✉rí❝✐♦ ❊✈❛♥❞r♦ ❊❧♦②✱ ♣❡❧♦ ❝♦♠♣r♦♠❡t✐❞♦ ❣r✉♣♦ ❞❡ ❡st✉❞♦✱ ♦s ♠♦♠❡♥t♦s ❞❡s❝♦♥tr❛í❞♦s ❡ ♣❡❧❛ ❛♠✐③❛❞❡ ✈❡r❞❛❞❡✐r❛✱ ❢♦rt❛❧❡❝✐❞❛ ❛♦ ❧♦♥❣♦ ❞♦ ❝✉rs♦✳

➚ t♦❞♦s ♦s ♣r♦❢❡ss♦r❡s ❡♥✈♦❧✈✐❞♦s ♥♦ ♣r♦❥❡t♦✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ❛q✉❡❧❡s q✉❡ ♠✐♥✐str❛✲ r❛♠ ❝✉rs♦s ♣❛r❛ ❛ t✉r♠❛ P❘❖❋▼❆❚ ✷✵✶✶✱ ❡♠ ❡s♣❡❝✐❛❧✱ à ♠✐♥❤❛ ♦r✐❡♥t❛❞♦r❛✱ Pr♦❢❛

❉r❛ _{❆❧✐❝❡ ❑✐♠✐❡ ▼✐✇❛ ▲✐❜❛r❞✐✱ ♣❡❧♦ ❛♣♦✐♦✱ ♣❛❝✐ê♥❝✐❛✱ ♣r♦✜ss✐♦♥❛❧✐s♠♦✱ ét✐❝❛ ❡ ❛t❡♥çã♦}

❘❡s✉♠♦

❖ ♦❜❥❡t✐✈♦ ♣r✐♥❝✐♣❛❧ ❞❡st❡ tr❛❜❛❧❤♦ é ♦ ❡st✉❞♦ ❞❛ ❝❛r❛❝t❡ríst✐❝❛ ❞❛ ❊✉❧❡r ❞❡ ♣♦❧✐❡✲ ❞r♦s✱ s✉♣❡r❢í❝✐❡s ❡ ❞❡ s♦♠❛ ❝♦♥❡①❛ ❞❡ s✉♣❡r❢í❝✐❡s✳ ➱ ♣r♦✈❛❞♦ q✉❡ s❡ ❞✉❛s s✉♣❡r❢í❝✐❡s t❡♠ ❛ ♠❡s♠❛ ❝❛r❛❝t❡ríst✐❝❛ ❞❡ ❊✉❧❡r✱ ❡♥tã♦ ❡❧❛s sã♦ ❤♦♠❡♦♠♦r❢❛s✳ ❆ r❡❝í♣r♦❝❛ é t❛♠❜é♠ ✈❡r❞❛❞❡✐r❛✱ ♣♦ré♠ s✉❛ ❞❡♠♦♥str❛çã♦ ❢♦❣❡ ❛♦ ❡s❝♦♣♦ ❞❡st❡ tr❛❜❛❧❤♦✳ P❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐✲ ♠❡♥t♦ ❞❛ ❛t✐✈✐❞❛❞❡ ♣❛r❛ ❛❧✉♥♦s ❞♦ ❡♥s✐♥♦ ♠é❞✐♦✱ ❢♦r❛♠ ❝♦♥str✉í❞♦s ♠❛t❡r✐❛✐s ❞✐❞át✐❝♦s ❝♦♠ ♦ ♦❜❥❡t✐✈♦ ❞❡ ♠♦t✐✈❛r ❡ ♠♦str❛r tr✐❛♥❣✉❧❛çõ❡s ❞❡ ❛❧❣✉♠❛s s✉♣❡r❢í❝✐❡s✱ ♥❡❝❡ssár✐❛s ♣❛r❛ ♦ ❝á❧❝✉❧♦ ❞❛s ❝❛r❛❝t❡ríst✐❝❛s ❞❡ ❊✉❧❡r✳

❆❜str❛❝t

❚❤❡ ♠❛✐♥ ❣♦❛❧ ♦❢ t❤✐s ✇♦r❦ ✐s t❤❡ st✉❞② ♦❢ t❤❡ ❊✉❧❡r ❝❤❛r❛❝t❡r✐st✐❝ ♦❢ ♣♦❧②❤❡❞r♦♥✱ s✉r❢❛❝❡s ❛♥❞ ♦❢ ❝♦♥♥❡❝t❡❞ s✉♠ ♦❢ s✉r❢❛❝❡s✳ ■t ✐s ❛❧s♦ ♣r♦✈❡❞ t❤❛t ✐❢ t✇♦ s✉r❢❛❝❡s ❤❛✈❡ t❤❡ s❛♠❡ ❊✉❧❡r ❝❤❛r❛❝t❡r✐st✐❝ t❤❡♥ t❤❡② ❛r❡ ❤♦♠❡♦♠♦r♣❤✐❝s✳ ❚❤❡ ❝♦♥✈❡rs❡ ✐s ❛❧s♦ tr✉❡✱ ❜✉t ✐t ✐s ♥♦t ♣r♦✈❡♥ ✐♥ t❤✐s ✇♦r❦✳ ❋♦r t❤❡ ❞❡✈❡❧♦♣♠❡♥t ♦❢ ❛♥ ❛❝t✐✈✐t② ❢♦r ❤✐❣❤ s❝❤♦♦❧ st✉❞❡♥ts ✇❡r❡ ♠❛❞❡ ❞✐❞❛❝t✐❝ ♠❛t❡r✐❛❧s ✐♥ ♦r❞❡r t♦ ♠♦t✐✈❛t❡ ❛♥❞ s❤♦✇ t❤❡ tr✐❛♥❣✉❧❛t✐♦♥s ♦❢ s♦♠❡ s✉r❢❛❝❡s✱ ♥❡❝❡ss❛r② t♦ ❝❛❧❝✉❧❛t❡ t❤❡ ❊✉❧❡r ❝❤❛r❛❝t❡r✐st✐❝s✳

❙✉♠ár✐♦

✶ ■♥tr♦❞✉çã♦ ✶✺

✷ Pr❡❧✐♠✐♥❛r❡s ✶✼

✷✳✶ ❊s♣❛ç♦s ▼étr✐❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✷✳✶✳✶ ❈♦♥t✐♥✉✐❞❛❞❡ ❡ ❍♦♠❡♦♠♦r✜s♠♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✷✳✶✳✷ Pr♦♣r✐❡❞❛❞❡s ❡❧❡♠❡♥t❛r❡s ❞❛s ❛♣❧✐❝❛çõ❡s ❝♦♥tí♥✉❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✷✳✶✳✸ ❍♦♠❡♦♠♦r✜s♠♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹

✸ ❈❛r❛❝t❡ríst✐❝❛ ❞❡ ❊✉❧❡r ✷✾

✸✳✶ ❙✉♣❡r❢í❝✐❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✸✳✶✳✶ ❙♦♠❛ ❈♦♥❡①❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸

✹ ❆t✐✈✐❞❛❞❡ r❡❧❛❝✐♦♥❛❞❛ ❛♦ ❊♥s✐♥♦ ▼é❞✐♦ ✸✾

✹✳✶ P❧❛♥♦ ❞❡ ❡♥s✐♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✹✳✶✳✶ ▼❡t♦❞♦❧♦❣✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✹✳✶✳✷ Pr♦❝❡ss♦ ❞❡ ❚r✐❛♥❣✉❧❛çã♦ ❞❡ ✉♠❛ s✉♣❡r❢í❝✐❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✹✳✶✳✸ ❯♠ ♣♦✉❝♦ ❞❛ ❤✐stór✐❛✲❊♥❝❡rr❛♠❡♥t♦ ❞❛ s❡①t❛ ❛✉❧❛ ✳ ✳ ✳ ✹✼

❘❡❢❡rê♥❝✐❛s ✹✾

✶ ■♥tr♦❞✉çã♦

❆ ❈❛r❛❝t❡ríst✐❝❛ ❞❡ ❊✉❧❡r é ✉♠ ❞♦s ♠❛✐s ✐♠♣♦rt❛♥t❡s ✐♥✈❛r✐❛♥t❡s t♦♣♦❧ó❣✐❝♦s✱ ❝♦♠ ❛ q✉❛❧ ♦❜t❡♠♦s ✉♠❛ ❝❧❛ss✐✜❝❛çã♦ ❞❛s s✉♣❡r❢í❝✐❡s ❝♦♠♣❛❝t❛s✱ ❝♦♥❡①❛s ❡ s❡♠ ❜♦r❞♦✳ ❙❡✉ ❡st✉❞♦ ❡♠ ár❡❛s ♠❛✐s ❡s♣❡❝í✜❝❛s ❞á ✉♠❛ ❝♦♥❞✐çã♦ à ❡①✐stê♥❝✐❛ ❞❡ ❝❛♠♣♦s ✈❡t♦r✐❛✐s ♥ã♦ ♥✉❧♦s✱ ♣♦ré♠ tr❛t❛✲s❡ ❞❡ ✉♠ ❛ss✉♥t♦ q✉❡ ♣♦❞❡ s❡r ❛❜♦r❞❛❞♦ ❡♠ ❞✐✈❡rs♦s ♥í✈❡✐s ❞❡ ❛♣r♦❢✉♥❞❛♠❡♥t♦✳ ◆♦ss♦ ♦❜❥❡t✐✈♦ é ❢❛③❡r ✉♠❛ ❛♣r❡s❡♥t❛çã♦ q✉❡ ♣♦ss❛ ♠♦t✐✈❛r ❞♦❝❡♥t❡s ❡ ❛❧✉♥♦s ❞♦ ❡♥s✐♥♦ ♠é❞✐♦ ❛ t❡r ✉♠❛ ✈✐sã♦ ♠❛✐s ❣❡r❛❧✱ ✐♥t❡r❡ss❛♥t❡✱ ♣♦ré♠ ❛❝❡ssí✈❡❧✳

▼✉✐t♦s ❧✐✈r♦s ❞✐❞át✐❝♦s ❛t✉❛❧♠❡♥t❡✱ ❧✐st❛❞♦s ♥♦ P◆▲❊▼✭❞❡ ✈ár✐♦s ❛♥♦s✮✱ tr❛③❡♠ ❛ss✉♥t♦s ♣r♦♥t♦s✱ ❞❡ ♠❛♥❡✐r❛ ❞✐r❡t❛ ❡ ❝♦♠ ♣♦✉❝♦ ❢✉♥❞❛♠❡♥t♦✱ ♥ã♦ ✐♥❝❡♥t✐✈❛♥❞♦ ❛ ♣❡♥s❛r ♦✉ ✐♠❛❣✐♥❛r ✉♠❛ ♣♦ss✐❜✐❧✐❞❛❞❡ ❞✐❢❡r❡♥t❡✳

❯♠ ❞❡ss❡s ❛ss✉♥t♦s✱ q✉❡ ❛♣r♦❢✉♥❞❛r❡♠♦s ♥❡ss❡ tr❛❜❛❧❤♦✱ tr❛t❛✲s❡ ❞❛ r❡❧❛çã♦ ❞❡ ❊✉❧❡r ♦✉ ❝♦♠♦ é ♠❛✐s ❝♦♥❤❡❝✐❞♦✱ ❚❡♦r❡♠❛ ❞❡ ❊✉❧❡r✳

❆❧❣✉♥s ❧✐✈r♦s ❞✐❞át✐❝♦s sã♦ ❜r❡✈❡s ❝♦♠ r❡s♣❡✐t♦ ❛♦ ❛ss✉♥t♦✳ ❉❡s❞❡ ✉♠ ✧✈❛♠♦s ❝❛❧❝✉❧❛r V −A+F ♣❛r❛ ✈❡r q✉❛♥t♦ ❞á✧ ❡ ❝♦♠♣❧❡t❛r t❛❜❡❧❛s ❝♦♠ ✈ár✐♦s ❡①❡♠♣❧♦s ❞❡ ♣♦❧✐❡❞r♦s✱ ❡♠ ♣❛rt✐❝✉❧❛r ❝♦♥✈❡①♦s✱ ♣❛r❛ ❢❛③❡r ❝♦♠ q✉❡ ♦ ❛❧✉♥♦ ♣❡r❝❡❜❛ q✉❡ ❛ r❡❧❛çã♦ ✈❛❧❡ ♣❛r❛ ♣♦❧✐❡❞r♦ ❝♦♥✈❡①♦s✳ ❍á ♦✉tr♦s q✉❡✱ ✐♥st✐❣❛♠ ♦ ❛❧✉♥♦ ❛ t❛♠❜é♠ ✈❡r✐✜❝❛r q✉❡ ❛ r❡❧❛çã♦ ♣♦❞❡ ❛té ✈❛❧❡r ♣❛r❛ ♣♦❧✐❡❞r♦s ♥ã♦✲❝♦♥✈❡①♦s✳ P♦ré♠✱ ❤á ❧✐✈r♦s q✉❡ ♦❜s❡r✈❛♠ q✉❡ ♥❡♠ t♦❞♦ ♣♦❧✐❡❞r♦ ♥ã♦✲❝♦♥✈❡①♦ s❡❣✉❡ ❛ r❡❧❛çã♦ ❞❡ ❊✉❧❡r✱ ♦✉ ❛té ❞❡♠♦♥str❛♠ ❞❡ ♠❛♥❡✐r❛ ❡❧❡❣❛♥t❡ ❡ ✐♥t❡❧✐❣í✈❡❧✳

❆ ❘❡❧❛çã♦✴❚❡♦r❡♠❛ ❞❡ ❊✉❧❡r t❡♠ s✐❞♦ ❡st✉❞❛❞❛ ❡ ❡♥s✐♥❛❞❛ ❤á ❞é❝❛❞❛s✱ ❞❡s❞❡ ❉❡s✲ ❝❛rt❡s ❛té ♦s ❞✐❛s ❛t✉❛✐s✱ ❞❡♠♦♥str❛♥❞♦ s❡✉ ❞❡s❛✜♦ ❡ s✉❛ ✐♠♣♦rtâ♥❝✐❛✱ ♦❜s❡r✈❛❞♦s ❡♠ ❞✐✈❡rs♦s tr❛❜❛❧❤♦s ♣❡r❞✐❞♦s ♥❛ ❤✐stór✐❛ ❡✱ ♣♦r ✜♠✱ s❡✉ ✈❡r❞❛❞❡✐r♦ s✐❣♥✐✜❝❛❞♦ ♥❛s ♠ã♦s ❞❡ P♦✐♥❝❛ré✳

◆♦ ❝❛♣ít✉❧♦ ✷✱ ❞❛r❡♠♦s ❜❛s❡ ♣❛r❛ ❡str✉t✉r❛ ❞❡ss❡ tr❛❜❛❧❤♦✱ ❞❡✜♥✐♥❞♦ ❡ ❡①❡♠♣❧✐✜✲ ❝❛♥❞♦ ❝♦♥❝❡✐t♦s ✐♠♣♦rt❛♥t❡s s♦❜r❡ ❡s♣❛ç♦s ♠étr✐❝♦s✱ s✉❜❡s♣❛ç♦s✱ ❜♦❧❛s ❛❜❡rt❛s ❡ ❡s❢❡✲ r❛s✳ ■♥tr♦❞✉③✐r❡♠♦s ♦ ❝♦♥❝❡✐t♦ ❞❡ ❝♦♥t✐♥✉✐❞❛❞❡ ❡ ❤♦♠❡♦♠♦r✜s♠♦ q✉❡ s✉st❡♥t❛rá t♦❞♦ ♦ tr❛❜❛❧❤♦✱ ♣♦✐s s❡rá ♦ t✐♣♦ ❞❡ ❞❡❢♦r♠❛çã♦ q✉❡ ✉s❛r❡♠♦s ❡♠ ♥♦ss♦ ♦❜❥❡t♦ ❞❡ ❡st✉❞♦✳ ❚❛♠❜é♠ s❡rã♦ ✈✐st♦s ❡s♣❛ç♦s t♦♣♦❧ó❣✐❝♦s ❡ s✉❛s ♣r♦♣r✐❡❞❛❞❡s✱ ♥♦ ✐♥t✉✐t♦ ❞❡ ❞❡✜♥✐r ❛ ❝❛r❛❝t❡ríst✐❝❛ ❞❡ ❊✉❧❡r ♣❛r❛ ❡s♣❛ç♦s ♠❛✐s ❣❡r❛✐s✳

◆♦ ❝❛♣ít✉❧♦ ✸ ✐♥✐❝✐❛r❡♠♦s ♦ ♦❜❥❡t♦ ❞❡ ♥♦ss♦ ❡st✉❞♦✱ ❞❡ tã♦ s✐♠♣❧❡s ❡ ❛té ♠❡s♠♦ ❛♣r❡s❡♥t❛❞♦ ♥♦ ❡♥s✐♥♦ ❢✉♥❞❛♠❡♥t❛❧✱ s❡ ♠♦str♦✉ ❢r✉tí❢❡r♦ ❡♠ ❡st✉❞♦s ♠❛✐s ❛♣r♦❢✉♥❞❛✲ ❞♦s✱ ♠❡r❡❝❡♥❞♦ s❡✉ ❞❡st❛q✉❡✳ ■♥tr♦❞✉③✐r❡♠♦s ♦ ♣r♦❝❡ss♦ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦s ❡❧❡♠❡♥t♦s

✶✻ ■♥tr♦❞✉çã♦

❞❛ ❢ór♠✉❧❛ ♣❛r❛ ✉♠❛ s✉♣❡r❢í❝✐❡ q✉❛❧q✉❡r✱ ✉s❛♥❞♦ ❛ ♥♦çã♦ ❞❡ tr✐❛♥❣✉❧❛çã♦✳ ◆❛ ú❧t✐♠❛ s❡çã♦✱ ✉t✐❧✐③❛r❡♠♦s s✉♣❡r❢í❝✐❡s ❝♦♥❤❡❝✐❞❛s ♣❛r❛ ❡❢❡t✉❛r ✉♠❛ ♦♣❡r❛çã♦✱ ❝❤❛♠❛❞❛ s♦♠❛ ❝♦♥❡①❛✱ ♣❛r❛ ♦❜t❡♥çã♦ ❞❡ ♥♦✈❛s s✉♣❡r❢í❝✐❡s✳ ❉❛r❡♠♦s ❞❡st❛q✉❡ ❛ três t❡♦r❡♠❛s✱ s❡♥❞♦ ♦ ♣r✐♥❝✐♣❛❧ ❞❡❧❡s ❛ ❝❧❛ss✐✜❝❛çã♦ ❞❡ s✉♣❡r❢í❝✐❡s ❢❡❝❤❛❞❛s✱ ✈✐❛ ❝❛r❛❝t❡ríst✐❝❛ ❞❡ ❊✉❧❡r✳

✷ Pr❡❧✐♠✐♥❛r❡s

❈♦♠♦ ♦ ♦❜❥❡t✐✈♦ ♣r✐♥❝✐♣❛❧ ❞❡st❡ tr❛❜❛❧❤♦ é ♦ ❡st✉❞♦ ❞❡ ✉♠ ✐♥✈❛r✐❛♥t❡ t♦♣♦❧ó❣✐❝♦✱ ♣r❡❝✐s❛♠♦s ✐♥tr♦❞✉③✐r ♦ ❝♦♥❝❡✐t♦ ❞❡ ❤♦♠❡♦♠♦r✜s♠♦✱ ✉♠❛ ❛♣❧✐❝❛çã♦ ❜✐❥❡t♦r❛✱ ❝♦♥tí♥✉❛ ❡ ❝♦♠ ✐♥✈❡rs❛ ❝♦♥tí♥✉❛✳ ❉❡ss❡ ♠♦❞♦✱ ♥❡st❡ ❝❛♣ít✉❧♦ ❛♣r❡s❡♥t❛♠♦s ❛❧❣✉♥s r❡s✉❧t❛❞♦s ♥❡❝❡ssár✐♦s ❛♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞♦ tr❛❜❛❧❤♦✳

✷✳✶ ❊s♣❛ç♦s ▼étr✐❝♦s

◆❛ ●❡♦♠❡tr✐❛ ❊✉❝❧✐❞✐❛♥❛✱ ❞❡✜♥❡✲s❡ ❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ ❞♦✐s ♣♦♥t♦s ❞✐st✐♥t♦s ❝♦♠♦ s❡♥❞♦ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞♦ s❡❣♠❡♥t♦ ❞❡t❡r♠✐♥❛❞♦ ♣♦r ❡❧❡s✳ P♦st❡r✐♦r♠❡♥t❡✱ ♥❛ ●❡♦✲ ♠❡tr✐❛ ❆♥❛❧ít✐❝❛✱ ✐♥tr♦❞✉③✐♥❞♦✲s❡ ✉♠ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s ❝❛rt❡s✐❛♥❛s ♦rt♦❣♦♥❛✐s✱ ❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ ♦s ♣♦♥t♦s ❞♦ ♣❧❛♥♦ ❝❛rt❡s✐❛♥♦ P = (a, b) ❡ Q = (c, d) é ❞❛❞❛ ♣♦r

d(P, Q) = (a−c)2_{+ (b−d)}2✳ ❆ ❞✐stâ♥❝✐❛ ❡♥tr❡ ❞♦✐s ♣♦♥t♦s ❞❡ ✉♠❛ ❝✐❞❛❞❡ ♣♦❞❡

♥ã♦ s❡r ♦ s❡❣♠❡♥t♦ ❞❡ r❡t❛ ❞❡t❡r♠✐♥❛❞♦ ♣♦r ❡❧❡s✱ ♣♦✐s ♠✉✐t❛s ✈❡③❡s ❡st❡ ❝❛♠✐♥❤♦ ♥ã♦ é ♣♦ssí✈❡❧✳ ❈❛♠✐♥❤❛♥❞♦ ♣❡❧❛s r✉❛s✱ ✈♦❝❡ t❡rá ✉♠❛ ♦✉tr❛ ♥♦çã♦ ❞❡ ❞✐stâ♥❝✐❛✳ ❊ s❡ ❞❡✜♥✐r♠♦s ❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ ❞♦✐s ♣♦♥t♦s✱ ❝♦♠♦ s❡♥❞♦ ✶✱ s❡ ❡❧❡s ❢♦r❡♠ ❞✐❢❡r❡♥t❡s ❡ ③❡r♦ s❡ ❢♦r❡♠ ✐❣✉❛✐s❄ ❖ q✉❡ t♦❞❛s ❡st❛s ❞❡✜♥✐çõ❡s t❡♠ ❡♠ ❝♦♠✉♠❄ ❆ r❡s♣♦st❛ é q✉❡ t♦✲ ❞❛s s❛t✐s❢❛③❡♠ ❛s ❝♦♥❞✐çõ❡s ❛❜❛✐①♦ q✉❡ ❞❡✜♥✐r❡♠♦s ❝♦♠♦ s❡♥❞♦ ✉♠❛ ♠étr✐❝❛ ♦✉ ✉♠❛ ❞✐stâ♥❝✐❛✳

❉❡✜♥✐çã♦ ✷✳✶✳ ❯♠❛ ♠étr✐❝❛ ♥✉♠ ❝♦♥❥✉♥t♦ M é ✉♠❛ ❢✉♥çã♦ d : M × M → R✱

q✉❡ ❛ss♦❝✐❛ ❛ ❝❛❞❛ ♣❛r ❞❡ ❡❧❡♠❡♥t♦s x, y ∈ M ✉♠ ♥ú♠❡r♦ r❡❛❧ d(x, y)✱ ❝❤❛♠❛❞♦ ❛

❞✐stâ♥❝✐❛ ❞❡x❛y✱ ❞❡ ♠♦❞♦ q✉❡ s❡❥❛♠ s❛t✐s❢❡✐t❛s ❛s s❡❣✉✐♥t❡s ❝♦♥❞✐çõ❡s ♣❛r❛ q✉❛✐sq✉❡r x, y, z ∈M✿

✐✮ d(x, y)0❡ d(x, y) = 0⇔x=y ✐✐✮ d(x, y) =d(y, x)

✐✐✐✮ d(x, z)d(x, y) +d(y, z)✳

❯♠ ❡s♣❛ç♦ ♠étr✐❝♦ é ✉♠ ♣❛r(M, d)✱ ♦♥❞❡ M é ✉♠ ❝♦♥❥✉♥t♦ ❡ dé ✉♠❛ ♠étr✐❝❛ ❡♠ M✳ ❖s ❡❧❡♠❡♥t♦s ❞❡ ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦ ♣♦❞❡♠ s❡r ❞❡ ♥❛t✉r❡③❛ ❜❛st❛♥t❡ ❛r❜✐trár✐❛✿ ♥ú♠❡r♦s✱ ♣♦♥t♦s✱ ✈❡t♦r❡s✱ ♠❛tr✐③❡s✱ ❢✉♥çõ❡s✱ ❝♦♥❥✉♥t♦s✱ ❡t❝✳

✶✽ Pr❡❧✐♠✐♥❛r❡s

❉❡✜♥✐çã♦ ✷✳✷✳ ❙❡❥❛ (M, d) ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦✳ ❚♦❞♦ s✉❜❝♦♥❥✉♥t♦ S ⊂ M ♣♦❞❡ s❡r ❝♦♥s✐❞❡r❛❞♦✱ ❞❡ ♠♦❞♦ ♥❛t✉r❛❧✱ ❝♦♠♦ ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦✿ ❜❛st❛ ❝♦♥s✐❞❡r❛r ❛ r❡str✐çã♦ ❞❡d❛S×S✱ ♦✉ s❡❥❛✱ ✉s❛r ♣❛r❛ ♦s ❡❧❡♠❡♥t♦s ❞❡ S❛ ♠❡s♠❛ ❞✐stâ♥❝✐❛ q✉❡ ❡❧❡s ♣♦ss✉í❛♠ ❝♦♠♦ ❡❧❡♠❡♥t♦s ❞❡ M✳ ◆❡ss❛s ❝♦♥❞✐çõ❡s✱ ❞✐③❡♠♦s q✉❡ S é ✉♠ s✉❜❡s♣❛ç♦ ❞❡ M ❡ ❛ ♠étr✐❝❛ ❞❡ S ❞✐③✲s❡ ✐♥❞✉③✐❞❛ ♣❡❧❛ ❞❡ M✳

❱❛♠♦s ❛♣r❡s❡♥t❛r ❛❧❣✉♥s ❡①❡♠♣❧♦s ❞❡ ♠étr✐❝❛s✳ ❆ ♣r✐♠❡✐r❛ ❞❡❧❛s é ❛ q✉❡ ❥á ♠❡♥✲ ❝✐♦♥❛♠♦s✱ ❛ ♠étr✐❝❛ ✧③❡r♦✲✉♠✧✳ ❉❛❞♦ M = ∅ ❞❡✜♥❡✲s❡ d : M × M → R ♣♦♥❞♦

d(x, x) = 0 ❡ d(x, y) = 1s❡x=y.P❡❧❛ ❞❡✜♥✐çã♦ ❞❡ ♠étr✐❝❛✱ ❛s ❝♦♥❞✐çõ❡s i, iisã♦ ✈❡r✐✲ ✜❝❛❞❛s ♣❡❧❛ ♣ró♣r✐❛ ❞❡✜♥✐çã♦ ❞❛ ♠étr✐❝❛ ✧③❡r♦✲✉♠✧ ❡ ♣❛r❛ ❛ ❝♦♥❞✐çã♦iii❝♦♥s✐❞❡r❡♠♦s ♣r✐♠❡✐r❛♠❡♥t❡x=y❡y=z✳ ❚❡♠♦s ❛ss✐♠d(x, y) = 1✱d(x, z) = 1❡d(y, z) = 0✳ ▲♦❣♦✱

♥❡st❡ ❝❛s♦✱ d(x, y) =d(x, z) +d(y, z)✳ ❙❡ x= y ❡ y=z t❡♠♦s✱ d(x, y) = 1✱ d(x, z) = 1

❡ d(y, z) = 1✱d(x, y)≤d(x, z) +d(y, z). ❖s ❞❡♠❛✐s ❝❛s♦s r❡❝❛❡♠ ♥❡st❡s✳

❯♠ ❡①❡♠♣❧♦ ✐♠♣♦rt❛♥t❡ ❞❡ ❡s♣❛ç♦ ♠étr✐❝♦ é ♦ ❡s♣❛ç♦ ❡✉❝❧✐❞✐❛♥♦ Rn✳ ❖s ♣♦♥t♦s

❞❡ Rn sã♦ ♥✲✉♣❧❛s _{x= (x1, ..., xn)} ♦♥❞❡ ❝❛❞❛ ✉♠❛ ❞❛s _n ❝♦♦r❞❡♥❛❞❛s _xi é ✉♠ ♥ú♠❡r♦

r❡❛❧✳ P♦❞❡♠♦s ❞❡✜♥✐r ❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ ❞♦✐s ♣♦♥t♦s ♥♦ Rn ❞❡ três ♠♦❞♦s✳ ❉❛❞♦s

x= (x1, ..., xn) ❡ y= (y1, ..., yn)✱ ❡s❝r❡✈❡♠♦s✿

d(x, y) =(x1−y1)2+· · ·+ (xn−yn)2 = n_i=1[(xi−yi)2]1/2 ,

d′

(x, y) = |x1−y1|+· · ·+|xn−yn|= n

i=1

|xi−yi|

❡

d′′

(x, y) = max{|x1−y1|, . . . ,|xn−yn|}= max

1in|xi−yi|.

❆s ❢✉♥çõ❡sd, d′ , d′′

:Rn_×Rn_→Rsã♦ ♠étr✐❝❛s✳ ❆ ♠étr✐❝❛_dé ❝❤❛♠❛❞❛_euclidiana✳

◆♦ ❝❛s♦ ♣❛rt✐❝✉❧❛r ❞♦ ❝♦♥❥✉♥t♦ R ❞♦s ♥ú♠❡r♦s r❡❛✐s ❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ ❞♦✐s ♣♦♥t♦s

x, y,∈Ré ❞❛❞❛ ♣♦r _{d(x, y) =|x−y|}✳ ❱❡r✐✜❝❛✲s❡ ✐♠❡❞✐❛t❛♠❡♥t❡ ❛s ❝♦♥❞✐çõ❡s ❞❡ _i à_iii

♣❡❧❛s ♣r♦♣r✐❡❞❛❞❡s ❡❧❡♠❡♥t❛r❡s ❞♦ ✈❛❧♦r ❛❜s♦❧✉t♦ ❞❡ ♥ú♠❡r♦s r❡❛✐s✳ ❊st❛ é ❛ ❝❤❛♠❛❞❛ ✧♠étr✐❝❛ ✉s✉❛❧✧✳

❉❡✜♥✐❞❛ ✉♠❛ ♠étr✐❝❛ ❡♠ ✉♠ ❝♦♥❥✉♥t♦ M✱ ✐st♦ é✱ t❡♥❞♦ ❛❣♦r❛ ❛ ♥♦çã♦ ❞❡ ❞✐stâ♥✲ ❝✐❛ ❡♥tr❡ ♦s ♣♦♥t♦s✱ ♦ ♣ró①✐♠♦ ♦❜❥❡t✐✈♦ é ❞❡✜♥✐r ❝♦♥t✐♥✉✐❞❛❞❡ ❞❡ ❢✉♥çõ❡s✳ P♦rt❛♥t♦✱ ✈❡r❡♠♦s ❛❧❣✉♥s ❡❧❡♠❡♥t♦s ❡ss❡♥❝✐❛✐s ♣❛r❛ s✉❛ ❞❡✜♥✐çã♦✳

❉❡✜♥✐çã♦ ✷✳✸✳ ❙❡❥❛♠ (M, d) ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦✱ a∈M ❡ r∈R∗

+✳

✐✮ ❆ ❜♦❧❛ ❛❜❡rt❛ ❞❡ ❝❡♥tr♦ a ❡ r❛✐♦r é ❞❡✜♥✐❞❛ ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✿ B(a, r) :={x∈M, d(x, a)< r},

❊s♣❛ç♦s ▼étr✐❝♦s ✶✾

✐✐✮ ❆ ❜♦❧❛ ❢❡❝❤❛❞❛ ♦✉ ❞✐s❝♦ ❞❡ ❝❡♥tr♦ a ❡ r❛✐♦ r é ❞❡✜♥✐❞♦ ♣♦r✿

D(a, r) :={x∈M, d(x, a)r},

♦✉ s❡❥❛✱ é ♦ ❝♦♥❥✉♥t♦ ❢♦r♠❛❞♦ ♣❡❧♦s ♣♦♥t♦s ❞❡ M q✉❡ ❡stã♦ ❛ ✉♠❛ ❞✐stâ♥❝✐❛ ♠❡♥♦r ❞♦ q✉❡ ♦✉ ✐❣✉❛❧ ❛ r ❞♦ ♣♦♥t♦ a✳

✐✐✐✮ ❆ ❡s❢❡r❛ ❞❡ ❝❡♥tr♦ a ❡ r❛✐♦r é ❞❡✜♥✐❞❛ ♣♦r✿

S(a, r) :={x∈M, d(x, a) =r}.

❖❜s❡r✈❡ q✉❡ t♦❞♦ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ♥♦r♠❛❞♦ E é ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦✱ ❝♦♠ ❛ ♠étr✐❝❛ d ♣r♦✈❡♥✐❡♥t❡ ❞❛ ♥♦r♠❛✱ ✐✳❡✳ d(x, y) = x−y ✳ ❊♥tã♦ ❡s❝r❡✈❡♠♦s✿

B(a, r) :={x∈E,|x−a|< r}

D(a, r) :={x∈E,|x−a|r}

S(a, r) :={x∈E,|x−a|=r}

❙❡ X é ✉♠ s✉❜❡s♣❛ç♦ ❞♦ ❡s♣❛ç♦ ♠étr✐❝♦ M✱ ❡♥tã♦ ♣❛r❛ ❝❛❞❛ a ∈X ❡ ❝❛❞❛ r >0✱

♣♦❞❡♠♦s ❞❡✜♥✐r ❛ BX(a, r)❛ ❜♦❧❛ ❛❜❡rt❛ ❞❡ ❝❡♥tr♦ a ❡ r❛✐♦r✱ r❡❧❛t✐✈❛♠❡♥t❡ à ♠étr✐❝❛

✐♥❞✉③✐❞❛ ❡♠X✳ ◆❡st❡ ❝❛s♦✱ t❡♠✲s❡BX(a, r) =B(a, r)X♦♥❞❡B(a, r)é ❛ ❜♦❧❛ ❛❜❡rt❛

❞❡ ❝❡♥tr♦ a ❡ r❛✐♦ r ♥♦ ❡s♣❛ç♦ M✳ ❆♥❛❧♦❣❛♠❡♥t❡ t❡♠♦s✿ DX(a, r) = D(a, r)X ❡

SX(a, r) = S(a, r)X✳

❆s ❜♦❧❛s ❛❜❡rt❛s ❞❡♣❡♥❞❡♠ ❞❛ ♠étr✐❝❛ q✉❡ s❡ ✉s❛✱ ♣♦r ❡①❡♠♣❧♦✱ s❡ M ❡stá ♠✉♥✐❞♦ ❞❛ ♠étr✐❝❛ ③❡r♦✲✉♠ ❡♥tã♦✱ ♣❛r❛ t♦❞♦ a ∈ M✱ t❡♠✲s❡ B(a, r) = D(a, r) = M s❡ r > 1

❡ B(a, r) =D(a, r) = a s❡ r < 1✳ P♦r ♦✉tr♦ ❧❛❞♦✱ B(a,1) =a ❡ D(a,1) =M✳ ❚❡♠✲s❡ t❛♠❜é♠ S(a, r) =∅ s❡r = 1 ❡ S(a,1) = M−a.

❈♦♠ ❛ ♠étr✐❝❛ ✉s✉❛❧ ❞❛ r❡t❛✱ ♣❛r❛ t♦❞♦ a ∈ R ❡ t♦❞♦ _{r > 0}✱ ❛ ❜♦❧❛ ❛❜❡rt❛ ❞❡

❝❡♥tr♦ a❡ r❛✐♦ r é ♦ ✐♥t❡r✈❛❧♦ ❛❜❡rt♦ (a−r, a+r)✱ ♣♦✐s ❛ ❝♦♥❞✐çã♦ |x−a|< r ❡q✉✐✈❛❧❡ a < x− a < r✱ ♦✉ s❡❥❛✿ a −r < x < a+r✳ ❆♥❛❧♦❣❛♠❡♥t❡✱ D(a, r) é ♦ ✐♥t❡r✈❛❧♦ [a−r, a+r]❡ ❛ ❡s❢❡r❛ S(a, r) t❡♠ ❛♣❡♥❛s ❞♦✐s ♣♦♥t♦s✿ a−r ❡ a+r✳

◆♦ ♣❧❛♥♦R2✱ ❛ ❜♦❧❛ ❛❜❡rt❛_{B(a, r)}é ♦ ✐♥t❡r✐♦r ❞❡ ✉♠ ❝ír❝✉❧♦ ❞❡ ❝❡♥tr♦ _a❡ r❛✐♦_r♦✉

♦ ✐♥t❡r✐♦r ❞❡ ✉♠ q✉❛❞r❛❞♦ ❞❡ ❝❡♥tr♦ a ❡ ❧❛❞♦s ❞❡ ❝♦♠♣r✐♠❡♥t♦s 2r✱ ♣❛r❛❧❡❧♦s ❛♦s ❡✐①♦s✱ ♦✉ ❡♥tã♦ ♦ ✐♥t❡r✐♦r ❞❡ ✉♠ q✉❛❞r❛❞♦ ❞❡ ❝❡♥tr♦ a ❡ ❞✐❛❣♦♥❛✐s ♣❛r❛❧❡❧❛s ❛♦s ❡✐①♦s✱ ❛♠❜❛s ❞❡ ❝♦♠♣r✐♠❡♥t♦s 2r✳ ❊st❡s ❝❛s♦s ❝♦rr❡s♣♦♥❞❡♠ ❛♦ ✉s❛r♠♦s ❡♠ R2 ❛s ♠étr✐❝❛s _d✱ _d′

♦✉ d′′ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❆ ❡s❢❡r❛

S(a, r) é ♦ ❜♦r❞♦ ❞❛ ✜❣✉r❛ ❝♦rr❡s♣♦♥❞❡♥t❡ ❡ D(a, r)✱

✷✵ Pr❡❧✐♠✐♥❛r❡s

✭❛✮ (x−a₁)2+ (y−a₂)2< r2✳ ✭❜✮|x−a₁|< r❡|y−a₂|<2✳ ✭❝✮ |x−a₁|+|y−a₂|< r✳

✷✳✶✳✶ ❈♦♥t✐♥✉✐❞❛❞❡ ❡ ❍♦♠❡♦♠♦r✜s♠♦s

❯♠ ❞♦s ♦❜❥❡t✐✈♦s ❞♦ tr❛❜❛❧❤♦ é ♠♦str❛r q✉❡ s❡ ❞✉❛s s✉♣❡r❢í❝✐❡s ♣♦ss✉❡♠ ❛ ♠❡s♠❛ ❝❛r❛❝t❡ríst✐❝❛ ❞❡ ❊✉❧❡r✱ ❡♥tã♦ ❡❧❛s sã♦ ❤♦♠❡♦♠♦r❢❛s✱ ♦✉ s❡❥❛✱ ❡①✐st❡ ✉♠ ❤♦♠❡♦♠♦r✜s♠♦ ❡♥tr❡ ❡❧❛s✳ ◆❛ r❡❛❧✐❞❛❞❡✱ ❡st❡ ❝♦♥❝❡✐t♦ é ✉t✐❧✐③❛❞♦ ❡♥tr❡ ❡s♣❛ç♦s t♦♣♦❧ó❣✐❝♦s✱ q✉❡ ✈❡r❡♠♦s ❛ s❡❣✉✐r✳ ❈♦♠❡ç❛♠♦s ❝♦♠ ❛ ❞❡✜♥✐çã♦ ❞❡ ❤♦♠❡♦♠♦r✜s♠♦s ❡♥tr❡ ❡s♣❛ç♦s ♠étr✐❝♦s✱ q✉❡ é ✉♠ ❝❛s♦ ♣❛rt✐❝✉❧❛r✱ ♠❛s ♥ã♦ ♠❡♥♦s ✐♠♣♦rt❛♥t❡✳

❉❡✜♥✐çã♦ ✷✳✹✳ ❙❡❥❛♠ M, N ❡s♣❛ç♦s ♠étr✐❝♦s✳ ❉✐③✲s❡ q✉❡ ❛ ❛♣❧✐❝❛çã♦ f : M → N é ❝♦♥tí♥✉❛ ♥♦ ♣♦♥t♦ a∈M q✉❛♥❞♦✱ ♣❛r❛ t♦❞♦ε >0❞❛❞♦✱ é ♣♦ssí✈❡❧ ♦❜t❡r δ >0t❛❧ q✉❡

d(x, a)< δ ✐♠♣❧✐❝❛ d(f(x), f(a))< ε✳

❉✐③✲s❡ q✉❡f :M →N é ❝♦♥tí♥✉❛ q✉❛♥❞♦ ❡❧❛ é ❝♦♥tí♥✉❛ ❡♠ t♦❞♦s ♦s ♣♦♥t♦sa∈M✳

❊q✉✐✈❛❧❡♥t❡♠❡♥t❡✱ f :M →N é ❝♦♥tí♥✉❛ ♥♦ ♣♦♥t♦ a ∈M q✉❛♥❞♦✱ ❞❛❞❛ q✉❛❧q✉❡r ❜♦❧❛ B′

= B(f(a);ε) ❞❡ ❝❡♥tr♦ f(a)✱ ♣♦❞❡✲s❡ ❡♥❝♦♥tr❛r ✉♠❛ ❜♦❧❛ B = B(a;δ), ❞❡ ❝❡♥tr♦ a✱ t❛❧ q✉❡ f(B)⊂B′✳

◆♦ ✐♠♣♦rt❛♥t❡ ❝❛s♦ ♣❛rt✐❝✉❧❛r ❡♠ q✉❡ M ⊂R❡_{f :M →}R✱ ❞✐③❡r q✉❡ _f é ❝♦♥tí♥✉❛

♥♦ ♣♦♥t♦ a ∈ M s✐❣♥✐✜❝❛ ❛✜r♠❛r q✉❡ ♣❛r❛ t♦❞♦ ε > 0✱ ❡①✐st❡ δ > 0 t❛❧ q✉❡ x ∈ M ❡ a −δ < x < a+δ ✐♠♣❧✐❝❛ f(a)−ε < f(x) < a+ε✳ ❖✉ s❡❥❛✱ f tr❛♥s❢♦r♠❛ ♦s ♣♦♥t♦s ❞❡ M q✉❡ ❡stã♦ ♥♦ ✐♥t❡r✈❛❧♦ ❛❜❡rt♦ (a−δ, a+δ)❡♠ ♣♦♥t♦s ❞♦ ✐♥t❡r✈❛❧♦ ❛❜❡rt♦ (f(a)−ε, f(a) +ε)✳

❊s♣❛ç♦s ▼étr✐❝♦s ✷✶

M ✉♠❛ ❜♦❧❛ B✱ ❞❡ ❝❡♥tr♦ a✱ t❛❧ q✉❡ f|B s❡❥❛ ❝♦♥tí♥✉❛ ♥♦ ♣♦♥t♦ a✱ ❡♥tã♦ f :M →N

é ❝♦♥tí♥✉❛ ♥♦ ♣♦♥t♦ a✳ ❙❡❣✉❡ ❞❛í q✉❡✱ s❡ ♣❛r❛ t♦❞❛ ♣❛rt❡ ❧✐♠✐t❛❞❛ X ⊂ M✱ f|X ❢♦r

❝♦♥tí♥✉❛✱ ❡♥tã♦ f :M →N é ❝♦♥tí♥✉❛✳

❊①❡♠♣❧♦ ✷✳✶✳ ❉❛❞❛ f : M → N✱ s✉♣♦♥❤❛♠♦s q✉❡ ❡①✐st❛ ✉♠❛ ❝♦♥st❛♥t❡ c > 0

✭❝❤❛♠❛❞❛ ❝♦♥st❛♥t❡ ❞❡ ▲✐♣s❝❤✐t③ ✮t❛❧ q✉❡d(f(x), f(y))≤cd(x, y)✱ q✉❛✐sq✉❡r q✉❡ s❡❥❛♠

x, y ∈M✳ ❉✐③❡♠♦s ❡♥tã♦ q✉❡f é ✉♠❛ ❛♣❧✐❝❛çã♦ ❧✐♣s❝❤✐t③✐❛♥❛✳ ◆❡st❡ ❝❛s♦✱ f é ❝♦♥tí♥✉❛✳ ❈♦♠ ❡❢❡✐t♦✱ ❞❛❞♦ ε > 0✱ t♦♠❡♠♦s δ = ε_c✳ ❊♥tã♦ d(x, a) < δ ⇒ d(f(x), f(a)) cd(x, a) < cδ =ε✳ ❙❡ c= 1✱ ❞✐③❡♠♦s q✉❡ f é ✉♠❛ ❝♦♥tr❛çã♦ ❢r❛❝❛✳ ❙ã♦ ❡①❡♠♣❧♦s ❞❡ ❝♦♥tr❛çõ❡s ❢r❛❝❛s✱ ♣♦rt❛♥t♦ ❝♦♥tí♥✉❛s✱ ❛ ♥♦r♠❛ || ||:E →R❡♠ ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧

♥♦r♠❛❞♦ E❀ ❛ ♣r♦❥❡çã♦ pi :M1× · · · ×Mn→ Mi ❞❡✜♥✐❞❛ ♣♦r pi(x1,· · ·xn) = xi ♣❛r❛

❝❛❞❛ i = 1,· · · , n ❡ ❛ ♦♣❡r❛çã♦ s♦♠❛ s : E ×E → E✱ s(x, y) = x+y ❡♠ ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ♥♦r♠❛❞♦ E✱ q✉❛♥❞♦ s❡ t♦♠❛ ❡♠ E×E ❛ ♥♦r♠❛ ||(x, y)||=|x|+|y|✳

❯♠❛ ❛♣❧✐❝❛çã♦ f : M → N ❝❤❛♠❛✲s❡ ❧♦❝❛❧♠❡♥t❡ ❧✐♣s❝❤✐t③✐❛♥❛✱ q✉❛♥❞♦ ❝❛❞❛ ♣♦♥t♦ a∈M é ❝❡♥tr♦ ❞❡ ✉♠❛ ❜♦❧❛ B =B(a, r)✱ t❛❧ q✉❡ ❛ r❡str✐çã♦ f|B é ❧✐♣s❝❤✐t③✐❛♥❛✳ ❯♠❛

❛♣❧✐❝❛çã♦ ❧♦❝❛❧♠❡♥t❡ ❧✐♣s❝❤✐t③✐❛♥❛ é✱ ❡✈✐❞❡♥t❡♠❡♥t❡✱ ❝♦♥tí♥✉❛✳

❊①❡♠♣❧♦ ✷✳✷✳ ❆ ❢✉♥çã♦ f : R _→ R✱ ❞❛❞❛ ♣♦r _{f(x) = x}n✱ ✭_n ✐♥t❡✐r♦ ♣♦s✐t✐✈♦✮é

❧✐♣s❝❤✐t③✐❛♥❛ ❡♠ ❝❛❞❛ ♣❛rt❡ ❧✐♠✐t❛❞❛ ❞❡ R✱ ♣♦✐s s❡ s❡ _|x|a ❡|y|a ❡♥tã♦|xn_−yn_|=

|x−y|·|xn−₁

+xn−₂

y+· · ·+yn−₁

||x−y|(||x|n−₁

+|x|n−₂

|y|+· · ·+|y|n−₁

|)c·|y−x|✱

♦♥❞❡ c = n·an−₁✳ ❙❡❣✉❡✲s❡ q✉❡ ✉♠ ♣♦❧✐♥ô♠✐♦

p(x) = a0 +a1x+· · ·+anxn ❝✉♠♣r❡

❛ ❝♦♥❞✐çã♦ ❞❡ ▲✐♣s❝❤✐t③ ❡♠ ❝❛❞❛ ✐♥t❡r✈❛❧♦ ❧✐♠✐t❛❞♦ [a, b]✳ ❈♦♥❝❧✉í♠♦s✱ ❡♠ ♣❛rt✐❝✉❧❛r✱

q✉❡ t♦❞♦ ♣♦❧✐♥ô♠✐♦ p : R _→ R é ✉♠❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛✳ ❉❡ ♠♦❞♦ ❛♥á❧♦❣♦✱ ♣♦❞❡♠♦s

♠♦str❛r q✉❡ ❛ ❢✉♥çã♦ r : R_{− {0} →} R✱ ❞❡✜♥✐❞❛ ♣♦r _{r(x) =} 1

x✱ é ❝♦♥tí♥✉❛✳ Pr♦✈❛♠♦s

♣r✐♠❡✐r♦ q✉❡✱ ♣❛r❛ ❝❛❞❛ k > 0✱ r é ❧✐♣s❝❤✐t③✐❛♥❛ ♥♦ ❝♦♥❥✉♥t♦ Xk ={x ∈R :|x| ≥k}✳

❖r❛✱ s❡ |x| ≥k ❡|y| ≥k✱ ❡♥tã♦|r(x)−r(y)|=|1

x −

1

y|= x−y

x·y ≤c· |x−y|✱ ♦♥❞❡ c=

1

✷✷ Pr❡❧✐♠✐♥❛r❡s

✷✳✶✳✷ Pr♦♣r✐❡❞❛❞❡s ❡❧❡♠❡♥t❛r❡s ❞❛s ❛♣❧✐❝❛çõ❡s ❝♦♥tí♥✉❛s

Pr♦♣♦s✐çã♦ ✷✳✺✳ ❆ ❝♦♠♣♦st❛ ❞❡ ❞✉❛s ❛♣❧✐❝❛çõ❡s ❝♦♥tí♥✉❛s é ❝♦♥tí♥✉❛✳ ▼❛✐s ♣r❡❝✐s❛✲ ♠❡♥t❡✱ s❡ f : M → N é ❝♦♥tí♥✉❛ ♥♦ ♣♦♥t♦ a ❡ g : N → P é ❝♦♥tí♥✉❛ ♥♦ ♣♦♥t♦ f(a)✱

❡♥tã♦ g◦f :M →P é ❝♦♥tí♥✉❛ ♥♦ ♣♦♥t♦ a✳ ❉❡♠♦♥str❛çã♦✿

❉❛❞♦ ε > 0✱ ❛ ❝♦♥t✐♥✉✐❞❛❞❡ ❞❡ g ♥♦ ♣♦♥t♦ f(a) ♥♦s ♣❡r♠✐t❡ ♦❜t❡r λ > 0 t❛❧ q✉❡

y∈N✱d(y, f(a))< λ⇒d(g(y), g(f(a))) < ε✳ P♦r s✉❛ ✈❡③✱ ❞❛❞♦ λ >0✱ ❛ ❝♦♥t✐♥✉✐❞❛❞❡

❞❡ f ♥♦ ♣♦♥t♦ a ♥♦s ❢♦r♥❡❝❡ δ > 0t❛❧ q✉❡ x∈M✱ d(x, a) < δ⇒ d(f(x), f(a))< λ ⇒

d(g(f(x)), g(f(a))) < ε✳

❈♦r♦❧ár✐♦ ✷✳✻✳ ❆ r❡str✐çã♦ ❞❡ ✉♠❛ ❛♣❧✐❝❛çã♦ ❝♦♥tí♥✉❛ é ❝♦♥tí♥✉❛✳ ▼❛✐s ❡①❛t❛♠❡♥t❡✱ s❡ f :M →N é ❝♦♥tí♥✉❛ ❡ X ⊂M✱ ❡♥tã♦ f|X :X →N é ❝♦♥tí♥✉❛✳

❉❡♠♦♥str❛çã♦✿ ❈♦♠ ❡❢❡✐t♦✱ f|X =f ◦i✱ ♦♥❞❡ i: X → M é ❛ ✐♥❝❧✉sã♦✱ ❞❡✜♥✐❞❛

♣♦r i(x) = x, x∈X✳

❊①❡♠♣❧♦ ✷✳✸✳ ❈♦♥t✐♥✉✐❞❛❞❡ ❞❛ ♠✉❧t✐♣❧✐❝❛çã♦✳ ❙❡❥❛ E ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ♥♦r♠❛❞♦✳ ❈♦♥s✐❞❡r❡ ❛ ❛♣❧✐❝❛çã♦ m:R_{×E →E}✱ ♦♥❞❡ _{m(λ, x) =λ·x}✳ ❙❡ _{[λ],[µ],[x],[y]}sã♦ _≤a

❡♥tã♦d[m(λ, x), m(µ, y)] = ||λ·x−µ·y||=||λ·x−µ·x+µ·x−µ·y|| ≤ |λ−µ| · ||x||+

|µ| · ||x−y|| ≤a(|λ−µ|+|x−y|) = a·d[(λ, x),(µ, y)]✳ ❙❡❣✉❡✲s❡ q✉❡ m é ❧✐♣s❝❤✐t③✐❛♥❛ ❡♠ ❝❛❞❛ ♣❛rt❡ ❧✐♠✐t❛❞❛ ❞❡ R_×E ❡✱ ♣♦r ❝♦♥s❡❣✉✐♥t❡✱ _m:R_{×E →E} é ❝♦♥tí♥✉❛✳ ❊♠

♣❛rt✐❝✉❧❛r✱ ❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❡ ♥ú♠❡r♦s r❡❛✐s✱ m : R_×R _→ R_{, m(x, y) = x·y}✱ é ✉♠❛

❢✉♥çã♦ ❝♦♥tí♥✉❛✳

❉❛❞♦s ♦s ❡s♣❛ç♦s ♠étr✐❝♦s M, N1 ❡ N2✱ ❞❛r ✉♠❛ ❛♣❧✐❝❛çã♦ f : M → N1 × N2

❡q✉✐✈❛❧❡ ❛ ✉♠ ♣❛r ❞❡ ❛♣❧✐❝❛çõ❡s f1 : M → N1 ❡ f2 : M → N2✱ ❝❤❛♠❛❞❛s ❛s ❢✉♥çõ❡s

❝♦♦r❞❡♥❛❞❛s ❞❡ f✱ t❛✐s q✉❡ f(x) = (f1(x), f2(x)) ♣❛r❛ t♦❞♦ x ∈ M✳ ❊s❝r❡✈❡✲s❡ f =

(f1, f2)✳ ❈♦♥s✐❞❡r❛♥❞♦✲s❡ ❛s ♣r♦❥❡çõ❡s p1 :N1×N2 →N1 ❡p2 :N1×N2 →N2 ✱ t❡♠✲s❡

f1 =p1◦f ❡ f2 = p2 ◦f✳ ❆ ♣r♦♣♦s✐çã♦ s❡❣✉✐♥t❡✱ ❡♠❜♦r❛ ❞❡ s✐♠♣❧❡s ❞❡♠♦♥str❛çã♦✱ é

❢✉♥❞❛♠❡♥t❛❧✳

Pr♦♣♦s✐çã♦ ✷✳✼✳ ❆ ❛♣❧✐❝❛çã♦ f : M → N1 ×N2 é ❝♦♥tí♥✉❛ ✭♥♦ ♣♦♥t♦ a ∈ M✮ s❡✱ ❡

s♦♠❡♥t❡ s❡✱ s✉❛s ❢✉♥çõ❡s ❝♦♦♦r❞❡♥❛❞❛s f1 :M →N1 ❡ f2 :M →N2 sã♦ ❝♦♥tí♥✉❛s ✭♥♦

❊s♣❛ç♦s ▼étr✐❝♦s ✷✸

❉❡♠♦♥str❛çã♦✿ ❙❡f é ❝♦♥tí♥✉❛ ❡♥tã♦ ♦ ♠❡s♠♦ ♦❝♦rr❡ ❝♦♠f1 =p1◦f ❡f2 =p2◦f

♣♦rq✉❡ ❛s ♣r♦❥❡çõ❡s p1 ❡p2 sã♦ ❝♦♥tí♥✉❛s✳ P❛r❛ ♣r♦✈❛r ❛ r❡❝í♣r♦❝❛✱ ✉s❛♠♦s ❡♠ N1×N2

❛ ♠étr✐❝❛ d[(x1, x2),(y1, y2)] ❂ ♠❛① {(x1, y1), d(x2, y2)}✳ ❉❛❞♦ ε > 0✱ ❝♦♠♦ f1 ❡ f2

sã♦ ❝♦♥tí♥✉❛s ♥♦ ♣♦♥t♦ a✱ ❡①✐st❡♠ δ1 ❡ δ2 t❛✐s q✉❡ d(x, a) < δ1 ⇒ d(f1(x), f1(a)) < ε

❡ d(x, a) < δ2 ⇒ d(f2(x), f2(a)) < ε✳ ❙❡❥❛ δ = ♠✐♥ {δ1, δ2}✳ ❊♥tã♦ d(x, a) < δ ⇒

d(f(x), f(a)) = ♠❛①{d(f1(x), f1(a)), d(f2(x), f2(a))}< ε✳ ▲♦❣♦f é ❝♦♥tí♥✉❛ ♥♦ ♣♦♥t♦

a✳

❈♦r♦❧ár✐♦ ✷✳✽✳ ❙❡ f1 : M1 → N1 ❡ f2 : M2 → N2 sã♦ ❝♦♥tí♥✉❛s✱ ❡♥tã♦ t❛♠❜é♠ é

❝♦♥tí♥✉❛ ❛ ❛♣❧✐❝❛çã♦

ϕ=f1×f2 :M1×M2 →N1×N2

❞❡✜♥✐❞❛ ♣♦r

ϕ(x1, x2) = (f1(x1), f2(x2)).

❈♦♠ ❡❢❡✐t♦✱ ❝♦♥s✐❞❡r❛♥❞♦ ❛s ♣r♦❥❡çõ❡s

p1 :M1×M2 →M1❡p2 :M1×M2 →M2

✈❡♠♦s q✉❡ ❛s ❢✉♥çõ❡s ❝♦♦r❞❡♥❛❞❛s ❞❡ ϕ sã♦

f1◦p1 :M1 ×M2 →N1❡f2◦p2 :M1×M2 →N2

❙❡❣✉❡ ❞❛ ♣r♦♣♦s✐çã♦ ❛♥t❡r✐♦r q✉❡ ϕ é ❝♦♥tí♥✉❛✳

Pr♦♣♦s✐çã♦ ✷✳✾✳ ❙❡❥❛♠ M ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦✱ E ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ♥♦r♠❛❞♦ ❡ f, g:

M → E, α, β :M → R ❛♣❧✐❝❛çõ❡s ❝♦♥tí♥✉❛s✱ ❝♦♠ _{β(x) = 0} ♣❛r❛ t♦❞♦ _{x ∈ M}✳ ❊♥tã♦

sã♦ ❝♦♥tí♥✉❛s ❛s ❛♣❧✐❝❛çõ❡s f+g :M → E, α·f : M →E ❡ α/β :M → R✱ ❞❡✜♥✐❞❛s

♣♦r

(f+g)(x) =f(x) +g(x),(α·f)(x) =α(x)·f(x),

❡

α β

(x) = α(x)

β(x).

❉❡♠♦♥str❛çã♦✿ ❋♦✐ ✈✐st♦ q✉❡ ❛s ❛♣❧✐❝❛çõ❡s r : R_{− {0} →} R✱ _{s : E×E → E} ❡

m :R_{×E →E}✱ ❞❛❞❛s ♣♦r _{r(x) = 1/x}✱_{s(x, y) = x+y}❡_{m(λ, x) = λ·x}✱ sã♦ ❝♦♥tí♥✉❛s✳

❈♦♥s✐❞❡r❡ ❛s s❡❣✉✐♥t❡s ❝♦♠♣♦s✐çõ❡s✿

M −→(f,g) E×E −→s E

x−→(f(x), g(x))→f(x) +g(x) f+g =s◦(f, g) ✭✷✳✶✮

M (−→α,f)R_{×E −→}m _E

✷✹ Pr❡❧✐♠✐♥❛r❡s

⎧ ⎨

⎩

M (−→α,β)R_×(R_{− {0})}(_−→id×r)R_×R_−→m R

x−→(α(x), β(x))−→α(x), 1

β(x)

→ α_β((_xx)₎

⎫ ⎬

⎭ α

β =m◦(id×r)◦(α, β) ✭✷✳✸✮ ❖s ❡sq✉❡♠❛s ❛❝✐♠❛ ✭♦♥❞❡ id :R_→Ré ❛ ❛♣❧✐❝❛çã♦ ✐❞❡♥t✐❞❛❞❡✮ ♠♦str❛♠ q✉❡ _f+g✱

α·f ❡α/β sã♦ ❝♦♥tí♥✉❛s✱ ❡♠ ✈✐rt✉❞❡ ❞❛s ♣r♦♣♦s✐çõ❡s ❛❝✐♠❛✳

❈♦r♦❧ár✐♦ ✷✳✶✵✳ ❙❡f, g :M →Rsã♦ ❢✉♥çõ❡s ❝♦♥tí♥✉❛s ❝♦♠ ✈❛❧♦r❡s r❡❛✐s✱ ❡♥tã♦ _f+g✱

f ·g ❡ ✭❝❛s♦ g(x)= 0 ♣❛r❛ t♦❞♦ x∈M✮ f /g sã♦ ❢✉♥çõ❡s ❝♦♥tí♥✉❛s✳

✷✳✶✳✸ ❍♦♠❡♦♠♦r✜s♠♦s

❉✐❢❡r❡♥t❡ ❞♦ q✉❡ ♦❝♦rr❡ ❡♠ ➪❧❣❡❜r❛ ▲✐♥❡❛r✱ ♦♥❞❡ ❛ ✐♥✈❡rs❛ ❞❡ ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ❧✐♥❡❛r ❜✐❥❡t✐✈❛ t❛♠❜é♠ é ❧✐♥❡❛r✱ ♦✉ ♥❛ ❚❡♦r✐❛ ❞♦ ●r✉♣♦s✱ ♦♥❞❡ ♦ ✐♥✈❡rs♦ ❞❡ ✉♠ ❤♦✲ ♠♦♠♦r✜s♠♦ ❜✐❥❡t✐✈♦ é ❛✐♥❞❛ ✉♠ ❤♦♠♦♠♦r✜s♠♦✱ ❡♠ ❚♦♣♦❧♦❣✐❛ ♦❝♦rr❡ ♦ ❢❡♥ô♠❡♥♦ ❞❡ ❡①✐st✐r❡♠ ❢✉♥çõ❡s ❝♦♥tí♥✉❛s ❜✐❥❡t✐✈❛s f :M →N t❛✐s q✉❡ f−1 _{:N →M é ❞❡s❝♦♥tí♥✉❛✳}

❖ s❡❣✉✐♥t❡ ❡①❡♠♣❧♦ é ✉♠ ❝❧áss✐❝♦ ❡①❡♠♣❧♦ ❞❡st❛ s✐t✉❛çã♦✳

❊①❡♠♣❧♦ ✷✳✹✳ ❙❡❥❛ S1 _{= {(x, y) ∈ R}2_;x2 _+y2 _{= 1} ♦ ❝ír❝✉❧♦ ✉♥✐tár✐♦ ❞♦ ♣❧❛♥♦}

❡✉❝❧✐❞✐❛♥♦✳ ❆ ❢✉♥çã♦ f : [0,2π) → S1_{✱ ❞❡✜♥✐❞❛ ♣♦r f(t) = (cost, sent) é ❜✐❥❡t♦r❛✱}

❡ é ❝♦♥tí♥✉❛✱ ♣♦✐s s✉❛s ❢✉♥çõ❡s ❝♦♦r❞❡♥❛❞❛s✱ cos ❡ sen ♦ sã♦✳ ❆ ❛♣❧✐❝❛çã♦ ✐♥✈❡rs❛✱ f−1 _:S1 _{→[0,2π) é ❞❡s❝♦♥tí♥✉❛ ♥♦ ♣♦♥t♦ P = (1,0)✳}

❉❡✜♥✐çã♦ ✷✳✶✶✳ ❙❡❥❛♠ M ❡N ❡s♣❛ç♦s ♠étr✐❝♦s✳ ❯♠ ❤♦♠❡♦♠♦r✜s♠♦ ❞❡ M s♦❜r❡ N é ✉♠❛ ❜✐❥❡çã♦ ❝♦♥tí♥✉❛ f : M → N ❝✉❥❛ ✐♥✈❡rs❛ f−1 _{: N → M t❛♠❜é♠ é ❝♦♥tí♥✉❛✳}

◆❡st❡ ❝❛s♦✱ ❞✐③✲s❡ q✉❡ M ❡ N sã♦ ❤♦♠❡♦♠♦r❢♦s✳

❙❡ f :M →N ❡ g :N →P sã♦ ❤♦♠❡♦♠♦r✜s♠♦s ❡♥tã♦ g◦f : M →P t❛♠❜é♠ é ✉♠ ❤♦♠❡♦♠♦r✜s♠♦✳

❊①❡♠♣❧♦ ✷✳✺✳ ❍♦♠❡♦♠♦r✜s♠♦ ❡♥tr❡ ❜♦❧❛s✳ ❙❡❥❛E ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ♥♦r♠❛❞♦✳ P❛r❛ t♦❞♦a ∈E ❡ ♣❛r❛ t♦❞♦ r❡❛❧λ= 0✱ ❛ tr❛♥s❧❛çã♦ ta:E →E ❡ ❛ ❤♦♠♦t❡t✐❛ mλ :E →E✱

❞❡✜♥✐❞❛s ♣♦r ta(x) = x+a ❡ mλ(x) = λ·x✱ sã♦ ❤♦♠❡♦♠♦r✜s♠♦s ❞❡ E✳ ❉❡ ❢❛t♦✱ ta ❡

mλ sã♦ ❝♦♥tí♥✉❛s ❡ ♣♦ss✉❡♠ ✐♥✈❡rs❛s✿ (t−a1 = t−a) ❡ (mλ)−1 = mµ, µ = 1/λ✱ ❛s q✉❛✐s

t❛♠❜é♠ sã♦ ❝♦♥tí♥✉❛s✳ ❉✉❛s ❜♦❧❛s ❛❜❡rt❛s B(a, r) ❡ B(b, s) ❡♠ E sã♦ ❤♦♠❡♦♠♦r❢❛s✳ ▼❛✐s ♣r❡❝✐s❛♠❡♥t❡✱ ❛ ❝♦♠♣♦st❛ϕ =tb◦ms/r◦t−a❞❡✜♥❡ ✉♠ ❤♦♠❡♦♠♦r✜s♠♦ϕ:E →E✳

P❛r❛ ❝❛❞❛ x∈E✱ t❡♠♦s ϕ(x) = b+s/r(x−a)✳ ■st♦ ♠♦str❛ q✉❡ ϕ ❝♦♥s✐st❡ ❡♠✿ ✶✳✮ ❚r❛♥s❧❛❞❛r B(a;r) ❞❡ ♠♦❞♦ ❛ ♣ôr s❡✉ ❝❡♥tr♦ ♥❛ ♦r✐❣❡♠❀

✷✳✮ ▼✉❧t✐♣❧✐❝❛r t♦❞♦s ♦s ✈❡t♦r❡s ♣♦r s/r ❞❡ ♠♦❞♦ q✉❡ ✈❡t♦r❡s ❞❡ ❝♦♠♣r✐♠❡♥t♦ < r ♣❛ss❡♠ ❛ t❡r ❝♦♠♣r✐♠❡♥t♦ < s✳ ■st♦ tr❛♥s❢♦r♠❛ B(0;r) ❡♠ B(0;s)❀

❊s♣❛ç♦s ▼étr✐❝♦s ✷✺

❆ss✐♠ ♦ ❤♦♠❡♦♠♦r✜s♠♦ ϕ :E →E é t❛❧ q✉❡ϕ(B(a;r)) =B(b;s)✳ ❉❛ ♠❡s♠❛ ♠❛♥❡✐r❛

s❡ ♠♦str❛ q✉❡ ❞✉❛s ❜♦❧❛s ❢❡❝❤❛❞❛s q✉❛✐sq✉❡r ❡♠ E sã♦ ❤♦♠❡♦♠♦r❢❛s✳ ❈♦♥✈é♠ ♦❜s❡r✈❛r q✉❡✱ ♥✉♠ ❡s♣❛ç♦ ♠étr✐❝♦ ❛r❜✐trár✐♦✱ ❞✉❛s ❜♦❧❛s ❛❜❡rt❛s ♣♦❞❡♠ ♥ã♦ s❡r ❤♦♠❡♦♠♦r❢❛s✳ ❖ ❡①❡♠♣❧♦ ♠❛✐s s✐♠♣❧❡s é ♦ ❞❡ ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦ M✱ q✉❡ ♣♦ss✉✐ ✉♠ ♣♦♥t♦ ✐s♦❧❛❞♦a❡ ✉♠ ♣♦♥t♦ ♥ã♦ ✐s♦❧❛❞♦ b✳ ❊①✐st❡ ✉♠ ❜♦❧❛s B(a;r) ={a}✱ ❛ q✉❛❧ ♥ã♦ ♣♦❞❡ s❡r ❤♦♠❡♦♠♦r❢❛

❛ ✉♠❛ ❜♦❧❛ ❛❜❡rt❛ ❞❡ ❝❡♥tr♦ b ♣♦✐s✱ ♣❛r❛ t♦❞♦ s >0✱ B(b;s) é ✉♠ ❝♦♥❥✉♥t♦ ✐♥✜♥✐t♦✳

❊①❡♠♣❧♦ ✷✳✻✳ ❆ ♣r♦❥❡çã♦ ❡st❡r❡♦❣rá✜❝❛✳ ❙❡❥❛♠ Sn_={x∈Rn+1_;x2

0+x21+x22+· · ·+

x2

n = 1} ❛ ❡s❢❡r❛ ✉♥✐tár✐❛ n✲❞✐♠❡♥s✐♦♥❛❧ ❡ p = (0,· · · ,0,1) ∈ Sn ♦ s❡✉ ♣ó❧♦ ♥♦rt❡✳

❆ ♣r♦❥❡çã♦ ❡st❡r❡♦❣rá✜❝❛ π : Sn _{− {p} → R}n _{❡st❛❜❡❧❡❝❡ ✉♠ ❤♦♠❡♦♠♦r✜s♠♦ ❡♥tr❡}

❛ ❡s❢❡r❛ ♠❡♥♦s ♦ ♣ó❧♦ ♥♦rt❡ ❡ ♦ ❡s♣❛ç♦ ❡✉❝❧✐❞✐❛♥♦ Rn✳ ●❡♦♠❡tr✐❝❛♠❡♥t❡✱ _π(x) é ♦

♣♦♥t♦ ❡♠ q✉❡ ❛ s❡♠✐✲r❡t❛ →

px ❡♥❝♦♥tr❛ ♦ ❤✐♣❡r♣❧❛♥♦ xn+1 = 0✱ q✉❡ ✐❞❡♥t✐✜❝❛♠♦s ❝♦♠

Rn✳ ❆ ✜♠ ❞❡ s❡ ♦❜t❡r ✉♠❛ ❢ór♠✉❧❛ ♣❛r❛ _π✱ ♦❜s❡r✈❡♠♦s q✉❡ ♦s ♣♦♥t♦s ❞❛ s❡♠✐✲r❡t❛

→

px tê♠ ❛ ❢♦r♠❛ p+t· (x−p)✱ ♦♥❞❡ t > 0✳ ❚❛❧ ♣♦♥t♦ ♣❡rt❡♥❝❡ ❛♦ ❤✐♣❡r♣❧❛♥♦ Rn

q✉❛♥❞♦ s✉❛ ú❧t✐♠❛ ❝♦♦r❞❡♥❛❞❛ 1 +t(xn+1 −1) é ③❡r♦✳ ❉❛í t✐r❛♠♦s t = 1/(1−xn+1)✳

❈♦♥✈❡❝✐♦♥❡♠♦s ♣♦r x′

= (x1,· · · , xn) q✉❛♥❞♦ x = (x1,· · · , xn, xn+1)✳ ❊♥tã♦✱ s❡♥❞♦

π(x) = p + (x −p)/(1−xn+1)✱ ♠♦str❛✲s❡ q✉❡ π(x) = x′/(1−xn+1)✳ ❆ ❡①♣r❡ssã♦

π(x) =x′

/(1−xn+1) ♠♦str❛ q✉❡ π :Sn− {p} →Rn é ❝♦♥tí♥✉❛✳

✭◆♦t❡ q✉❡ x ∈ Sn_{− {p} ❡①❝❧✉✐ ❛ ♣♦ss✐❜✐❧✐❞❛❞❡ x}

n+1 = 1✮✳ P❛r❛ ✈❡r✐✜❝❛r q✉❡ π é

✉♠ ❤♦♠❡♦♠♦r✜s♠♦✱ ❜❛st❛ ❝♦♥s✐❞❡r❛r ❛ ❛♣❧✐❝❛çã♦ ϕ : Rn _{→ S}n_{− {p}}✱ ❞❡✜♥✐❞❛ ♣♦r

ϕ(y) = x✱ ♦♥❞❡✱ ♥❛ ♥♦t❛çã♦ ❛❝✐♠❛✱ x′

= 2y/(|y|2 _{+ 1) ❡ x}

n+1 = (|y|2 −1)/(|y|2 + 1)✳

❈♦♥st❛t❛✲s❡ s❡♠ ❞✐✜❝✉❧❞❛❞❡ q✉❡ ϕ(π(x)) = x′ ♣❛r❛ t♦❞♦

x ∈Sn_{− {p} ❡ π(ϕ(y)) = y}

✷✻ Pr❡❧✐♠✐♥❛r❡s

❖s ❡❧❡♠❡♥t♦s ♠❛✐s ✐♠♣♦rt❛♥t❡s ❞❡ ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦ sã♦ ♦s ❝♦♥❥✉♥t♦s ❛❜❡rt♦s✱ q✉❡ ❞❛rã♦ ❛ ♠♦t✐✈❛çã♦ ♣❛r❛ ❛ ❞❡✜♥✐çã♦ ❞❡ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦✳

❉❡✜♥✐çã♦ ✷✳✶✷✳ ❙❡❥❛ X ✉♠ s✉❜❝♦♥❥✉♥t♦ ❞❡ ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦ M✳ ❯♠ ♣♦♥t♦ a∈X ❞✐③✲s❡ ✉♠ ♣♦♥t♦ ✐♥t❡r✐♦r ❛ X q✉❛♥❞♦ é ❝❡♥tr♦ ❞❡ ✉♠ ❜♦❧❛ ❛❜❡rt❛ ❝♦♥t✐❞❛ ❡♠ X✱ ♦✉ s❡❥❛✱ q✉❛♥❞♦ ❡①✐st❡ r >0 t❛❧ q✉❡ d(x, a)< r ⇒x∈X✳ ❈❤❛♠❛✲s❡ ♦ ✐♥t❡r✐♦r ❞❡ X ❡♠ M ❛♦ ❝♦♥❥✉♥t♦ ✐♥tX ❢♦r♠❛❞♦ ♣❡❧♦s ♣♦♥t♦s ✐♥t❡r✐♦r❡s ❛ X✳

❯♠ s✉❜❝♦♥❥✉♥t♦ ❞❡ A ❞❡ ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦ M ❞✐③✲s❡ ❛❜❡rt♦ ❡♠M q✉❛♥❞♦ t♦❞♦s ♦s s❡✉s ♣♦♥t♦s sã♦ ✐♥t❡r✐♦r❡s✱ ✐st♦ é✱ ✐♥t A=A✳ P❛r❛ ♣r♦✈❛r q✉❡ ✉♠ ❝♦♥❥✉♥t♦ A ⊂M é ❛❜❡rt♦ ❡♠ M✱ ❞❡✈❡♠♦s ♦❜t❡r✱ ♣❛r❛ ❝❛❞❛ x∈A✱ ✉♠ r❛✐♦ r > 0t❛❧ q✉❡ B(x;r)⊂A✳ Pr♦♣♦s✐çã♦ ✷✳✶✸✳ ❙❡❥❛ U ❛ ❝♦❧❡çã♦ ❞♦s s✉❜❝♦♥❥✉♥t♦s ❛❜❡rt♦s ❞❡ ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦

M✳ ❊♥tã♦✿

✭✶✮ M ∈U ❡_{∅ ∈}U✳ ✭❖ ❡s♣❛ç♦ ✐♥t❡✐r♦ ❡ ♦ ❝♦♥❥✉♥t♦ ✈❛③✐♦ sã♦ ❛❜❡rt♦s✳✮

✭✷✮ ❙❡A1,· · · , An ∈U ❡♥tã♦ A1∩ · · · ∩An ∈U✳ ✭❆ ✐♥t❡rs❡çã♦ ❞❡ ✉♠ ♥ú♠❡r♦ ✜♥✐t♦ ❞❡

❝♦♥❥✉♥t♦s ❛❜❡rt♦s é ✉♠ ❝♦♥❥✉♥t♦ ❛❜❡rt♦✮✳

✭✸✮ ❙❡ Aλ ∈ U ♣❛r❛ t♦❞♦ λ ∈ L ❡♥tã♦ A= _λ∈LAλ ∈ U✳ ✭❆ r❡✉♥✐ã♦ ❞❡ ✉♠❛ ❢❛♠í❧✐❛

q✉❛❧q✉❡r ❞❡ ❝♦♥❥✉♥t♦s ❛❜❡rt♦s é ✉♠ ❝♦♥❥✉♥t♦ ❛❜❡rt♦✳✮ ❉❡♠♦♥str❛çã♦✿

✶✮ ❖ ❡s♣❛ç♦ ♠étr✐❝♦ M é ❡✈✐❞❡♥t❡♠❡♥t❡✱ ❛❜❡rt♦ ❡♠ M✳ ■st♦ ♠♦str❛ ❝♦♠♦ ❛ ♣r♦♣r✐❡✲ ❞❛❞❡ ✧X é ❛❜❡rt♦✧ é r❡❧❛t✐✈❛✱ ✐st♦ é✱ ❞❡♣❡♥❞❡ ❞♦ ❡s♣❛ç♦ M ❡♠ q✉❡ s❡ ❝♦♥s✐❞❡r❛ X ✐♠❡rs♦✿ X é s❡♠♣r❡ ❛❜❡rt♦ ♥♦ ♣ró♣r✐♦ ❡s♣❛ç♦ X✳ P❛r❛ ✉♠ ❡①❡♠♣❧♦ ♠❡♥♦s tr✐✈✐❛❧✱ ♦❜s❡r✈❡♠♦s q✉❡ X = [0,1) é ✉♠ s✉❜❝♦♥❥✉♥t♦ ❛❜❡rt♦ ❞♦ ❡s♣❛ç♦ M = [0,1] : ❜❛st❛

♥♦t❛r q✉❡ ❝❛❞❛ ✐♥t❡r✈❛❧♦ ❞♦ t✐♣♦ [0, ε)✱ ❝♦♠ 0< ε1✱ é ❛❜❡rt♦ ♥❛ r❡t❛ R✳ ❚❛♠❜é♠

♦ ✐♥t❡r✈❛❧♦ ❛❜❡rt♦ (0,1)❞♦ ❡✐①♦ ❞❛s ❛❜s❝✐ss❛s ❡♠ R2 é ❛❜❡rt♦ ♥❡ss❡ ❡✐①♦ ♠❛s ♥ã♦ é

❛❜❡rt♦ ❡♠ R2✳ ❯♠ ❝♦♥❥✉♥t♦ q✉❡ é ❛❜❡rt♦ ❡♠ q✉❛❧q✉❡r ❡s♣❛ç♦ ♠étr✐❝♦ q✉❡ ♦ ❝♦♥t❡✲

♥❤❛ é ♦ ❝♦♥❥✉♥t♦ ✈❛③✐♦ ∅✳ ❈♦♠ ❡❢❡✐t♦✱ ♣❛r❛ ♣r♦✈❛r q✉❡ ✉♠ ❝♦♥❥✉♥t♦ X ♥ã♦ é ❛❜❡rt♦✱ ❞❡✈❡✲s❡ ❡①✐❜✐r ✉♠ ♣♦♥t♦ x ∈ X q✉❡ ♥ã♦ s❡❥❛ ✐♥t❡r✐♦r ❛ X✳ ■st♦ é ❡✈✐❞❡♥t❡♠❡♥t❡ ✐♠♣♦ssí✈❡❧ ❞❡ ❢❛③❡r q✉❛♥❞♦ X=∅✳ ▲♦❣♦ ∅ é ❛❜❡rt♦✳

✷✮ ❙✉♣♦♥❤❛♠♦s q✉❡ a ∈ A1,· · · , a ∈ An✳ ❈♦♠♦ ❡st❡s ❝♦♥❥✉♥t♦s sã♦ ❛❜❡rt♦s✱ ❡①✐st❡♠

r1 > 0,· · · , rn > 0 t❛✐s q✉❡ B(a, r1) ⊂ A1,· · · , B(a, rn) ⊂An✳ ❙❡❥❛ r ♦ ♠❡♥♦r ❞♦s

♥ú♠❡r♦s r1,· · · , rn✳ ❊♥tã♦

B(a;r)⊂B(a;r1)⊂A1,· · · , B(a;r)⊂B(a;rn)⊂An

❡ ❞❛í

B(a;r)⊂A1∩ · · · ∩An

❊s♣❛ç♦s ▼étr✐❝♦s ✷✼

✸✮ ❙❡❥❛ a ∈A✳ ❊♥tã♦✱ ❡①✐st❡ ✉♠ í♥❞✐❝❡ λ ∈ L t❛❧ q✉❡ a ∈ Aλ✳ ❈♦♠♦ ❡st❡ ❝♦♥❥✉♥t♦ é

❛❜❡rt♦✱ ❤á ✉♠❛ ❜♦❧❛B(a;r) ❝♦♥t✐❞❛ ❡♠ Aλ✳ ▲♦❣♦ B(a;r)⊂A✳

❱❛♠♦s ❛♣r❡s❡♥t❛r ❛❣♦r❛ ✉♠❛ ❝❛r❛❝t❡r✐③❛çã♦ ❞❡ ❢✉♥çõ❡s ❝♦♥tí♥✉❛s ✉s❛♥❞♦ ❝♦♥❥✉♥t♦s ❛❜❡rt♦s✳

Pr♦♣♦s✐çã♦ ✷✳✶✹✳ ❙❡❥❛ f :M −→N ✉♠❛ ❛♣❧✐❝❛çã♦ ❡♥tr❡ ❡s♣❛ç♦s ♠étr✐❝♦s✳ ❊♥tã♦ f é ❝♦♥tí♥✉❛ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❛ ✐♠❛❣❡♠ ✐♥✈❡rs❛ f−1_{(A)❞❡ q✉❛❧q✉❡r ❛❜❡rt♦A ❞❡N é ✉♠}

s✉❜❝♦♥❥✉♥t♦ ❛❜❡rt♦ ❞❡ M✳

❉❡♠♦♥str❛çã♦ ✷✳✶✺✳ ❙✉♣♦♥❤❛♠♦s f ❝♦♥tí♥✉❛ ❡ ❝♦♥s✐❞❡r❡♠♦s A ✉♠ s✉❜❝♦♥❥✉♥t♦ ❛❜❡rt♦ q✉❛❧q✉❡r ❞❡N✳ P❛r❛ ❝❛❞❛a∈f−1_{(A)✱ t❡♠✲s❡f(a)∈A✱ q✉❡ ♣♦r s❡r ❛❜❡rt♦✱ ❝♦♥✲}

té♠ ✉♠❛ ❜♦❧❛ ❛❜❡rt❛ ❝♦♠ ❝❡♥tr♦ f(a) ❡ r❛✐♦r✳ ❉❛ ❝♦♥t✐♥✉✐❞❛❞❡ ❞❡ f✱ ❡①✐st❡B(a, s)⊂

M t❛❧ q✉❡ f(B(a, s)) ⊂ B(f(a), r)) ⊂ A✳ P♦rt❛♥t♦✱ B(a, s) ⊂ f−1_{(B(f(a), r)) ⊂}

f−1_{(A) ❡ f}−1_{(A) é ❛❜❡rt♦ ❞❡ M✳ ❘❡❝✐♣r♦❝❛♠❡♥t❡✱ s❡❥❛ a ∈ M q✉❛❧q✉❡r✳ ❉❛❞❛}

B(f(a), ε)✱ q✉❡ é ✉♠ ❝♦♥❥✉♥t♦ ❛❜❡rt♦✱ ♣♦r ❤✐♣ót❡s❡✱ ❛ ✐♠❛❣❡♠ ✐♥✈❡rs❛ f−1_{(B(f(a), ε)é}

❛❜❡rt♦ ❡♠ M ❡ ❝♦♥té♠ a✳ P♦rt❛♥t♦✱ ❡①✐st❡ ✉♠❛ ❜♦❧❛ ❛❜❡rt❛ B(a, δ)⊂f−1_{(B(f(a), ε).}

❙❡❣✉❡ q✉❡ f(B(a, δ))⊂B(f(a), ε)), ♦✉ s❡❥❛ f é ❝♦♥tí♥✉❛✳

❊ss❛ ♣r♦♣♦s✐çã♦ ♠♦t✐✈❛✲♥♦s ❛ ❞❡✜♥✐r ✉♠ ❡s♣❛ç♦ ♠❛✐s ❣❡r❛❧✱ ♦♥❞❡ ♣❛r❛ s❡ ❢❛❧❛r ❞❡ ♣r♦①✐♠✐❞❛❞❡✱ ♥ã♦ ❤á ♥❡❝❡ss✐❞❛❞❡ ❞♦ ❝♦♥❝❡✐t♦ ❞❡ ✉♠❛ ♠étr✐❝❛✳

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

✭✶✮ X ∈τ ❡∅ ∈τ✳

✭✷✮ ❙❡ A1,· · · , An∈τ ❡♥tã♦ A1∩ · · · ∩An ∈τ✳

✭✸✮ ❙❡ Aλ ∈τ ♣❛r❛ t♦❞♦ λ∈L ❡♥tã♦ A=

λ∈LAλ ∈τ✳

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

❖❜s❡r✈❡ q✉❡ ❞❡st❛ ❢♦r♠❛✱ ✉♠❛ ❢✉♥çã♦ f : X −→ Y✱ ❡♥tr❡ ❡s♣❛ç♦s t♦♣♦❧ó❣✐❝♦s é ❝♦♥tí♥✉❛ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❛ ✐♠❛❣❡♠ ✐♥✈❡rs❛ ❞❡ q✉❛❧q✉❡r ❛❜❡rt♦ ❞❡ Y é ✉♠ ❛❜❡rt♦ ❞❡ X✳ ❚r❛❜❛❧❤❛r❡♠♦s ❝♦♠ ❡s♣❛ç♦s t♦♣♦❧ó❣✐❝♦s ❡s♣❡❝✐❛✐s✱ ❛ s❛❜❡r ❡s♣❛ç♦s ❞❡ ❍❛✉s❞♦r✛✱ ❝♦♥❡①♦s✱ ❝♦♠♣❛❝t♦s ❡ q✉♦❝✐❡♥t❡s✱ ❝✉❥❛s ❞❡✜♥✐çõ❡s ❛♣r❡s❡♥t❛♠♦s ❛ s❡❣✉✐r✳

❉❡✜♥✐çã♦ ✷✳✶✼✳ ❯♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦Xé ❞❡ ❍❛✉s❞♦r✛ s❡✱ ❞❛❞♦s ❞♦✐s ♣♦♥t♦s ❞✐st✐♥t♦s a ❡ b ❞❡ X✱ ❡①✐st❡♠ ❛❜❡rt♦s ❞✐s❥✉♥t♦s A ❝♦♥t❡♥❞♦ a ❡ B ❝♦♥t❡♥❞♦ b✳

❖❜s❡r✈❡ q✉❡ t♦❞♦ ❡s♣❛ç♦ ♠étr✐❝♦ é ❞❡ ❍❛✉s❞♦r✛✳

✷✽ Pr❡❧✐♠✐♥❛r❡s

❉❡✜♥✐çã♦ ✷✳✶✾✳ ❯♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ X é ❝♦♠♣❛❝t♦ s❡ t♦❞❛ ❝♦❜❡rt✉r❛ ❛❜❡rt❛ ❞❡ X✱ ✐✳❡✳ s❡ t♦❞❛ ❢❛♠í❧✐❛ ❞❡ ❛❜❡rt♦s ❝✉❥❛ r❡✉♥✐ã♦ é X✱ ❛❞♠✐t❡ ✉♠❛ s✉❜❝♦❜❡rt✉r❛ ✜♥✐t❛✱ ♦✉ s❡❥❛✱ ❡①✐st❡ ✉♠ ♥ú♠❡r♦ ✜♥✐t♦ ❞❡ ❛❜❡rt♦s ❞❡st❛ ❢❛♠í❧✐❛ ❝✉❥❛ r❡✉♥✐ã♦ é ❛✐♥❞❛ ♦ X✳

❙✉♣♦♥❤❛ ❛❣♦r❛ q✉❡ t❡♥❤❛♠♦s A ✉♠ ❝♦♥❥✉♥t♦ q✉❛❧q✉❡r ❡ f :X −→A ✉♠❛ ❢✉♥çã♦ ♦♥❞❡ X é ✉♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦✳ P♦❞❡♠♦s ❞❡✜♥✐r ❡♠ A ✉♠❛ t♦♣♦❧♦❣✐❛✱ ❝❤❛♠❛❞❛ t♦♣♦❧♦❣✐❛ q✉♦❝✐❡♥t❡✱ ❞❡✜♥✐♥❞♦✲s❡ q✉❡ s✉❜❝♦♥❥✉♥t♦s U sã♦ ❛❜❡rt♦s ❞❡ A s❡✱ ❡ s♦♠❡♥t❡ s❡✱f−1_{(U)✱ sã♦ ❛❜❡rt♦s ❞❡ X✳}

✸ ❈❛r❛❝t❡ríst✐❝❛ ❞❡ ❊✉❧❡r

❉❡♥tr❡ ♦s ✐♥✈❛r✐❛♥t❡s t♦♣♦❧ó❣✐❝♦s✿ ❝♦♥❡①ã♦✱ ❣r✉♣♦ ❢✉♥❞❛♠❡♥t❛❧✱ ❣r✉♣♦s ❞❡ ❤♦♠♦✲ ❧♦❣✐❛ s✐♠♣❧✐❝✐❛❧ ❡ ❝❛r❛❝t❡ríst✐❝❛ ❞❡ ❊✉❧❡r✱ ❡s❝♦❧❤❡♠♦s ♦ ú❧t✐♠♦ ♣❛r❛ ❝♦♥❢❡❝çã♦ ❞❡st❡ tr❛❜❛❧❤♦✳ ❘❡ss❛❧t❛♠♦s q✉❡ ❛ ❝❛r❛❝t❡ríst✐❝❛ ❞❡ ❊✉❧❡r ❞❡s❝r❡✈❡ ❡ ❝❧❛ss✐✜❝❛ ✐♠♣♦rt❛♥t❡s ❡s♣❛ç♦s t♦♣♦❧ó❣✐❝♦s✱ ❛tr❛✈és ❛♣❡♥❛s ❞❡ ✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦✳

❉❡✜♥✐çã♦ ✸✳✶✳ ❙❡❥❛ P ✉♠ ♣♦❧✐❡❞r♦ ❡ ❞❡♥♦t❡♠♦s ♣♦r ✈✱ ♦ ♥ú♠❡r♦ ❞❡ ✈ért✐❝❡✱ ❢ ♦ ♥ú♠❡r♦ ❞❡ ❢❛❝❡s ❡ ❡ ♦ ♥ú♠❡r♦ ❞❡ ❛r❡st❛s ❞❡ P✳ ❖ ♥ú♠❡r♦ χ(P) =v−e+f é ❝❤❛♠❛❞♦ ❞❡ ❝❛r❛❝t❡ríst✐❝❛ ❞❡ ❊✉❧❡r ❞❡ P✳

❊①❡♠♣❧♦ ✸✳✶✳ ❖ ♥ú♠❡r♦ ❞❡ ✈ért✐❝❡s✱ ❛r❡st❛s✱ ❢❛❝❡s ❡ ❛ ❝❛r❛❝t❡ríst✐❝❛ ❞❡ ❊✉❧❡r ❞♦s s❡❣✉✐♥t❡s ♣♦❧✐❡❞r♦s✿

✭❞✮ ❋✐❣✉r❛ ✶ ✭❡✮ ❋✐❣✉r❛ ✷ ✭❢✮ ❋✐❣✉r❛ ✸

❞✮ V = 6, a= 12, f = 8, χ= 2

❞✮ V = 14, a= 24, f = 14, χ= 4

❞✮ V = 20, a= 40, f = 20, χ= 0

✸✳✶ ❙✉♣❡r❢í❝✐❡s

❉❡✜♥✐çã♦ ✸✳✷✳ ❯♠❛ s✉♣❡r❢í❝✐❡ é ✉♠ ❡s♣❛ç♦ ❞❡ ❍❛✉s❞♦r✛✱ ❝♦♠♣❛❝t♦ ❡ ❝♦♥❡①♦✱ t❛❧ q✉❡ ♣❛r❛ ❝❛❞❛ ♣♦♥t♦✱ ❡①✐st❡ ✉♠ ❛❜❡rt♦ q✉❡ ♦ ❝♦♥té♠ ❡ q✉❡ é ❤♦♠❡♦♠♦r❢♦ ❛ ✉♠❛ ❜♦❧❛ ❛❜❡rt❛ ❞♦ R2✳

✸✵ ❈❛r❛❝t❡ríst✐❝❛ ❞❡ ❊✉❧❡r

◆❛ s❡q✉ê♥❝✐❛ ✐r❡♠♦s ❞❡t❡r♠✐♥❛r ❛ ❝❛r❛❝t❡ríst✐❝❛ ❞❡ ❊✉❧❡r ❞❡ s✉♣❡r❢í❝✐❡s✳ ◆❡st❡ ♣r♦❝❡ss♦✱ ❝♦♠♦ ❞❡♣❡♥❞❡♠♦s ❞♦ ♥ú♠❡r♦s ❞❡ ✈ért✐❝❡s✱ ❛r❡st❛s ❡ ❢❛❝❡s✱ ❡❢❡t✉❛r❡♠♦s ♦ ❝á❧❝✉❧♦ ❝♦♥s✐❞❡r❛♥❞♦ ✉♠❛ tr✐❛♥❣✉❧❛çã♦ ❞❛ s✉♣❡r❢í❝✐❡ ❡ ❛♣❧✐❝❛r❡♠♦s ❛ ❢ór♠✉❧❛✳ ❯♠❛ s✉♣❡r❢í❝✐❡ é tr✐❛♥❣✉❧á✈❡❧✭♥ã♦ é ❢á❝✐❧ ❞❡ ♣r♦✈❛r✮✱ q✉❛♥❞♦ ♣♦❞❡♠♦s ❞❡❝♦♠♣ô✲❧❛ ❡♠ ✉♠ ♥ú♠❡r♦ ✜♥✐t♦ ❞❡ ✈ért✐❝❡s✱ ❢❛❝❡s ❡ ❛r❡st❛s✳ P♦r ❡①❡♠♣❧♦✱ ♣♦❞❡♠♦s tr✐❛♥❣✉❧❛r ❛ ❡s❢❡r❛ ❡♠ q✉❛tr♦ ✈ért✐❝❡s✱ s❡✐s ❛r❡st❛s ❡ ✹ tr✐â♥❣✉❧♦s ✭❛r❡st❛s ❡ tr✐â♥❣✉❧♦s ❝✉r✈♦s✮✱ ❞❡ ♠♦❞♦ q✉❡ ♦ ♠♦❞❡❧♦ ❝♦rr❡s♣♦♥❞❡♥t❡ é ✉♠ t❡tr❛❡❞r♦✳ ➚s ✈❡③❡s✱ é ♠❛✐s ❢á❝✐❧ ❞❡❝♦♠♣♦r ❡♠ ♣♦❧í❣♦♥♦s✱ ♠❛s s❡ ♣✉❞❡r♠♦s ❞❡❝♦♠♣♦r ❡♠ ♣♦❧í❣♦♥♦s✱ ♣♦❞❡♠♦s ❞❡❝♦♠♣♦r ❡♠ tr✐â♥❣✉❧♦s✳ ❊ss❛ ❞❡❝♦♠♣♦s✐çã♦ s❡rá ❝❤❛♠❛❞❛ ❞❡ tr✐â♥❣✉❧❛çã♦✳ ❯♠❛ tr✐â♥❣✉❧❛çã♦ t❡♠ ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿

• ◗✉❛❧q✉❡r ❛r❡st❛ é ❛r❡st❛ ❞❡ ❡①❛t❛♠❡♥t❡ ❞♦✐s tr✐â♥❣✉❧♦s❀

• ❉❛❞♦s ❞♦✐s tr✐â♥❣✉❧♦s✱ ❡①✐st❡ ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ tr✐â♥❣✉❧♦s ❝♦♠❡ç❛♥❞♦ ❡♠ ✉♠

❞❡❧❡s ❡ t❡r♠✐♥❛♥❞♦ ♥♦ ♦✉tr♦✱ ❞❡ ♠♦❞♦ q✉❡ q✉❛✐sq✉❡r ❞♦✐s t❡r♠♦s ❝♦♥s❡❝✉t✐✈♦s ❞❡ss❛ s❡q✉ê♥❝✐❛ t❡♠ ✉♠❛ ❛r❡st❛ ❡♠ ❝♦♠✉♠✳

❚é❝♥✐❝❛s ❞❡ ❚♦♣♦❧♦❣✐❛ ❆❧❣é❜r✐❝❛ sã♦ ✉s❛❞❛s ♣❛r❛ ♣r♦✈❛r q✉❡ ❛ ❝❛r❛❝t❡ríst✐❝❛ ❞❡ ❊✉❧❡r ♥ã♦ ❞❡♣❡♥❞❡ ❞❛ tr✐❛♥❣✉❧❛çã♦✳

❊①❡♠♣❧♦ ✸✳✷✳ ❱❛♠♦s ❛♣r❡s❡♥t❛r ❛❧❣✉♥s ❡①❡♠♣❧♦s ❞❡ s✉♣❡r❢í❝✐❡s✳ ❉❡♥♦t❡♠♦s ♣♦r I ♦ ✐♥t❡r✈❛❧♦ ❢❡❝❤❛❞♦ [0,1]✳

✶✳ ❈✐❧✐♥❞r♦✿ é ♦ ❡s♣❛ç♦ q✉♦❝✐❡♥t❡ C = I×_I

∼ ✱ ♦♥❞❡ ” ∼ ” é ❛ r❡❧❛çã♦ ❞❡✜♥✐❞❛ ♣♦r

(x,0)∼(x,1)✳

✷✳ ❚♦r♦✿ é ♦ ❡s♣❛ç♦ q✉♦❝✐❡♥t❡ T2 ₌ I×_I

∼ ✱ ♦♥❞❡ ” ∼ ” é ❛ r❡❧❛çã♦ ❞❡✜♥✐❞❛ ♣♦r

(x,0)∼(x,1)❡ (0, y)∼(1, y)✳

✸✳ ❋❛✐①❛ ❞❡ ▼♦❡❜✐✉s✿ é ♦ ❡s♣❛ç♦ q✉♦❝✐❡♥t❡ C = I×_I

∼ ✱ ♦♥❞❡”∼”é ❛ r❡❧❛çã♦ ❞❡✜♥✐❞❛

♣♦r (x,0)∼(1,1−x)✳

✹✳ ●❛rr❛❢❛ ❞❡ ❑❧❡✐♥✿ é ♦ ❡s♣❛ç♦ q✉♦❝✐❡♥t❡ KB = I×_I

∼ ✱ ♦♥❞❡ ” ∼ ” é ❛ r❡❧❛çã♦

❞❡✜♥✐❞❛ ♣♦r (x,0)∼(x,1)❡ (0, y)∼(1,1−y)✳

✺✳ P❧❛♥♦ Pr♦❥❡t✐✈♦✿ é ♦ ❡s♣❛ç♦ q✉♦❝✐❡♥t❡ P2 _=S2_{/∼ ♦♥❞❡ x∼ −x✳}

■♥✐❝✐❛r❡♠♦s ♥❡st❡ ❝❛♣ít✉❧♦✱ ♦ ♣r♦❝❡ss♦ ❞❡ tr✐❛♥❣✉❧❛çã♦ ❞❡ ❛❧❣✉♠❛s ❞❛s ♣r✐♥❝✐♣❛✐s s✉♣❡r❢í❝✐❡s ❡ ❝❛❧❝✉❧❛r❡♠♦s s✉❛s r❡s♣❡❝t✐✈❛s ❝❛r❛❝t❡ríst✐❝❛s ❞❡ ❊✉❧❡r✿

❙✉♣❡r❢í❝✐❡s ✸✶

❱ért✐❝❡✿ 6,❆r❡st❛s✿ 12,❋❛❝❡s✿ 6

χ(C) = v−a+f χ(C) = 6−12 + 6

χ(C) = 0

✷✮ ❆ ❡s❢❡r❛ S2_✳

✭❣✮ ❚r✐❛♥❣✉❧❛çã♦ ❊s❢❡r❛ ✭❤✮ ❊s❢❡r❛

❱ért✐❝❡✿ 4,❆r❡st❛s✿ 6,❋❛❝❡s✿4

χ(S2) =v−a+f χ(S2) = 4−6 + 4

χ(S2) = 2

✸✮ ❋❛✐①❛ ❞❡ ▼♦❡❜✐✉s

✭✐✮ ❚r✐❛♥❣✉❧❛çã♦ ❋❛✐①❛ ❞❡ ▼♦❡❜✐✉s ✭❥✮ ❋❛✐①❛ ❞❡ ▼♦❡❜✐✉s

✸✷ ❈❛r❛❝t❡ríst✐❝❛ ❞❡ ❊✉❧❡r

χ(M) = v−a+f χ(M) = 6−12 + 6

χ(M) = 0

✹✮ ❖ ❚♦r♦ T2

✭❦✮ ❚r✐❛♥❣✉❧❛çã♦ ❚♦r♦ ✭❧✮ ❚♦r♦

❱ért✐❝❡✿ 9,❆r❡st❛s✿ 27,❋❛❝❡s✿ 18

χ(T2) =v−a+f χ(T2_{) = 9−27 + 18}

χ(T2) = 0

✺✮ ❆ ●❛rr❛❢❛ ❞❡ ❑❧❡✐♥

✭♠✮ ❚r✐❛♥❣✉❧❛çã♦ ●❛rr❛❞❛ ❞❡ ❑❧❡✐♥ ✭♥✮ ●❛rr❛❢❛ ❞❡ ❑❧❡✐♥

❱ért✐❝❡✿ 9,❆r❡st❛s✿ 27,❋❛❝❡s✿ 18

❙✉♣❡r❢í❝✐❡s ✸✸

χ(KB) = 9−27 + 18

χ(KB) = 0

✻✮ ❖ P❧❛♥♦ Pr♦❥❡t✐✈♦

✭♦✮ ❚r✐❛♥❣✉❧❛çã♦ P❧❛♥♦ Pr♦❥❡t✐✈♦ ✭♣✮ P❧❛♥♦ Pr♦❥❡t✐✈♦

❱ért✐❝❡✿ 6,❆r❡st❛s✿ 15,❋❛❝❡s✿10

χ(P2) =v−a+f χ(P2) = 6−15 + 10

χ(P2) = 1

❆ ❝❛r❛❝t❡ríst✐❝❛ ❞❡ ❊✉❧❡r ♥ã♦ ❞❡♣❡♥❞❡ ❞❛ tr✐❛♥❣✉❧❛çã♦✱ ❞❡♣❡♥❞❡♥❞♦ ❛♣❡♥❛s ❞❛ s✉✲ ♣❡r❢í❝✐❡ ❛ s❡r ❛♥❛❧✐s❛❞❛✳ ❖✉tr❛s s✉♣❡r❢í❝✐❡s s❡rã♦ ❝❛❧❝✉❧❛❞❛s ❛tr❛✈és ❞♦s ❡①❡♠♣❧♦s ❞❛❞♦s✱ t♦♠❛♥❞♦ ✉♠❛ ♦♣❡r❛çã♦ ❝❤❛♠❛❞❛ s♦♠❛ ❝♦♥❡①❛✳

✸✳✶✳✶ ❙♦♠❛ ❈♦♥❡①❛

❉❡ ♠❛♥❡✐r❛ ✐♥t✉✐t✐✈❛✱ ❛ s♦♠❛ ❝♦♥❡①❛ ❞❡ ❞✉❛s s✉♣❡r❢í❝✐❡s S1 ❡S2 é ❛ s✉♣❡r❢í❝✐❡S1♯S2

✸✹ ❈❛r❛❝t❡ríst✐❝❛ ❞❡ ❊✉❧❡r

❉❡✜♥✐çã♦ ✸✳✸✳ ❙❡❥❛♠ S1 ❡ S2 ❞✉❛s s✉♣❡r❢í❝✐❡s✱ ❝♦♠♣❛❝t❛s ❡ s❡♠ ❜♦r❞♦✳ ❊s❝♦❧❤❡♠♦s

D1 ⊂ S1 ❡ D2 ⊂ S2✱ s✉❜❝♦♥❥✉♥t♦s ❤♦♠❡♦♠♦r❢♦s ❛♦ ❞✐s❝♦ D2 ❡ s❡❥❛♠ h1 : D1 →D2 ❡

h2 : D2 → D2✱ ♦s r❡s♣❡❝t✐✈♦s ❤♦♠❡♦♠♦r✜s♠♦s✳ ❉❡✜♥✐♠♦s ❛ s♦♠❛ ❝♦♥❡①❛ ❞❡ S1 ❡ S2✱

❡ ❞❡♥♦t❛♠♦s ♣♦r S1♯S2✱ ❝♦♠♦ s❡♥❞♦ ♦ ❝♦♥❥✉♥t♦

(S1−intD1)∪(S2−intD2) ∼

♦♥❞❡ ❛ r❡❧❛çã♦ x∼y é ❞❛❞❛ ♣♦r✿

✐✮ ❙❡ x, y ❡stã♦ ♥♦ ❝♦♠♣❧❡♠❡♥t❛r ❞❡ ∂D1∪∂D2 ❡♥tã♦ x∼y⇔x=y❀

✐✐✮ ❈❛s♦ ❝♦♥trár✐♦✱ x∼y⇔h1(x) =h2(y)✳

❆ s♦♠❛ ❝♦♥❡①❛ ♥ã♦ ❞❡♣❡♥❞❡ ❞❛ ❡s❝♦❧❤❛ ❞♦s s✉❜❝♦♥❥✉♥t♦s D1 ❡D2✱ é ✉♠❛ s✉♣❡r❢í❝✐❡✱

❝♦♠ ❛ t♦♣♦❧♦❣✐❛ q✉♦❝✐❡♥t❡✳

❊①❡♠♣❧♦ ✸✳✸✳ ❊s❝r❡✈❡♥❞♦ S1 = S2 = T2 ❡♥tã♦ S1♯S2 = T2♯T2 é ❞❛❞❛ ♣❡❧❛ ✜❣✉r❛

❛❜❛✐①♦✿

❝♦♥❡①❛✶✳❥♣❣

Pr♦♣♦s✐çã♦ ✸✳✹✳ ❙❡❥❛♠ S1 ❡ S2 ❞✉❛s s✉♣❡r❢í❝✐❡s ❢❡❝❤❛❞❛s ✭❝♦♠♣❛❝t❛s ❡ s❡♠ ❜♦r❞♦✮✳

❊♥tã♦ χ(S1♯S2) =χ(S1) +χ(S2)−2✳

❉❡♠♦♥str❛çã♦✿

❉❡♥♦t❡♠♦s ♣♦r K1 ❡ K2 tr✐❛♥❣✉❧❛çõ❡s ❞❡ S1 ❡S2✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❙❡❥❛♠ χ(S1) =

v1 −e1 +f1 ❡ χ(S2) = v2 −e2 +f2✳ K

′

1 = K1 − △{a0a1a2} é ✉♠❛ tr✐❛♥❣✉❧❛çã♦ ❞❡

S1 −intD2✱ ♦♥❞❡ {a0a1a2} sã♦ ♦s ✈ért✐❝❡s ❞❡ ✉♠ tr✐â♥❣✉❧♦ ❡ K

′

2 = K2 − △{b0b1b2}

é ✉♠❛ tr✐❛♥❣✉❧❛çã♦ ❞❡ S2 − intD2✱ ♦♥❞❡ {b0b1b2} sã♦ ♦s ✈ért✐❝❡s ❞❡ ✉♠ tr✐â♥❣✉❧♦✳

❚♦♠❡♠♦s T = T

′

1∪T

′

2

∼ ✱ ♦♥❞❡ ai ∼ bi✱ i = 0,1,2 ❡ aiaj ∼ bibj✱ i, j = 0,1,2✳ ❈♦♠♦ T é

✉♠❛ tr✐❛♥❣✉❧❛çã♦ ♣❛r❛ χ(S1♯S2✮✱ ❡♥tã♦

χ(S1♯S2) = (v1+v2−3)−(e1+e2−3) + (f1+f2 −2) = χ(S1) +χ(S2)−2.

P❛r❛ ❡❢❡t✉❛r♠♦s ♦ ❝á❧❝✉❧♦ ❞❛ ❝❛r❛❝t❡ríst✐❝❛ ❞❡ ❊✉❧❡r ❞❡ s✉♣❡r❢í❝✐❡s ❢❡❝❤❛❞❛s ❛♣❧✐✲ ❝❛r❡♠♦s ♦ s❡❣✉✐♥t❡ t❡♦r❡♠❛✱ q✉❡ ❝❧❛ss✐✜❝❛ ❛s s✉♣❡r❢í❝✐❡s ♣♦r ❤♦♠❡♦♠♦r✜s♠♦s✳