❯♥✐✈❡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) = p(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)6d(x, y) +d(y, z)✳
❯♠ ❡s♣❛ç♦ ♠étr✐❝♦ é ✉♠ ♣❛r(M, d)✱ ♦♥❞❡M é ✉♠ ❝♦♥❥✉♥t♦ ❡ d é ✉♠❛ ♠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 6= ∅ ❞❡✜♥❡✲s❡ d : M × M → R ♣♦♥❞♦
d(x, x) = 0 ❡ d(x, y) = 1s❡x6=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❡x6=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=6 y ❡y 6=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) =p(x1−y1)2+· · ·+ (xn−yn)2 = Pni=1[(xi−yi)2]1/2 ,
d′
(x, y) = |x1−y1|+· · ·+|xn−yn|= n
X
i=1
|xi−yi|
❡
d′′
(x, y) = max{|x1−y1|, . . . ,|xn−yn|}= max
16i6n|xi−yi|.
❆s ❢✉♥çõ❡s d, 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 ❞❡ iàiii
♣❡❧❛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)6r},
♦✉ 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) =kx−yk✳ ❊♥tã♦ ❡s❝r❡✈❡♠♦s✿
B(a, r) :={x∈E,|x−a|< r}
D(a, r) := {x∈E,|x−a|6r}
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)TX♦♥❞❡B(a, r)é ❛ ❜♦❧❛ ❛❜❡rt❛
❞❡ ❝❡♥tr♦ a ❡ r❛✐♦ r ♥♦ ❡s♣❛ç♦ M✳ ❆♥❛❧♦❣❛♠❡♥t❡ t❡♠♦s✿ DX(a, r) = D(a, r)TX ❡ SX(a, r) = S(a, r)TX✳
❆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= 16 ❡ 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♦s2r✱ ♣❛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❡rδ >0 t❛❧ 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 = εc✳ ❊♥tã♦ d(x, a) < δ ⇒ d(f(x), f(a)) 6
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) = xn✱ ✭n ✐♥t❡✐r♦ ♣♦s✐t✐✈♦✮ é
❧✐♣s❝❤✐t③✐❛♥❛ ❡♠ ❝❛❞❛ ♣❛rt❡ ❧✐♠✐t❛❞❛ ❞❡R✱ ♣♦✐s s❡ s❡|x|6a❡|y|6a❡♥tã♦|xn−yn|=
|x−y|·|xn−1
+xn−2
y+· · ·+yn−1
|6|x−y|(||x|n−1
+|x|n−2
|y|+· · ·+|y|n−1
|)6c·|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
k2✳
✷✷ 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 > 0 t❛❧ 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❡sp1 :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❛❧✳
❊s♣❛ç♦s ▼étr✐❝♦s ✷✸
❉❡♠♦♥str❛çã♦✿ ❙❡f é ❝♦♥tí♥✉❛ ❡♥tã♦ ♦ ♠❡s♠♦ ♦❝♦rr❡ ❝♦♠f1 =p1◦f ❡f2 =p2◦f ♣♦rq✉❡ ❛s ♣r♦❥❡çõ❡sp1 ❡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)6= 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 ♣♦rr(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 x7−→(f(x), g(x))7→f(x) +g(x)
)
f+g =s◦(f, g) ✭✷✳✶✮
(
M (−→α,f)R×E −→m E
x7−→(α(x), f(x))7→α(x)·f(x)
)
✷✹ Pr❡❧✐♠✐♥❛r❡s
M (−→α,β)R×(R− {0})(−→id×r)R×R−→m R
x7−→(α(x), β(x))7−→α(x), 1
β(x)
7→ αβ((xx))
α
β =m◦(id×r)◦(α, β) ✭✷✳✸✮ ❖s ❡sq✉❡♠❛s ❛❝✐♠❛ ✭♦♥❞❡ id:R→R é ❛ ❛♣❧✐❝❛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)6= 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✐✈❛sf :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) ∈ R2;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❡❛❧λ6= 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❡♠♦sϕ(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 →Eé 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❡ ✉♠ ❜♦❧❛sB(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;x20+x21+x22+· · ·+
x2n = 1} ❛ ❡s❢❡r❛ ✉♥✐tár✐❛ n✲❞✐♠❡♥s✐♦♥❛❧ ❡ p = (0,· · · ,0,1) ∈ Sn ♦ s❡✉ ♣ó❧♦ ♥♦rt❡✳
❆ ♣r♦❥❡çã♦ ❡st❡r❡♦❣rá✜❝❛ π : Sn − {p} → Rn ❡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♥✉❛✳
P
x
y
π
(y)
π
(x)
n
R
✭◆♦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 → Sn − {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 > 0t❛❧ 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 =Sλ∈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< ε61✱ 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=
S
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✐✈♦
a
4a
4a
3a
3a
5a
5a1
a2
a0
✭♦✮ ❚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❡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♦sD1 ❡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✳