▼ô❝ ❧ô❝ ▼ô❝ ❧ô❝ ▼ë ➤➬✉ ✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳ ✶ ✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✷ ❈❤➢➡♥❣ ✶✳ ➜Þ♥❤ ❧ý ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ➳♥❤ ①➵ ❝♦ rót tr➟♥ ❦❤➠♥❣ ❣✐❛♥ ♠❡tr✐❝ ♥ã♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✶ ❑❤➠♥❣ ❣✐❛♥ ♠❡tr✐❝ ♥ã♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✷ ➜Þ♥❤ ❧ý ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ➳♥❤ ①➵ ❝♦ rót ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ❈❤➢➡♥❣ ✷✳ ❈➳❝ tÝ♥❤ ❝❤✃t t➠♣➠ ❝ñ❛ ❦❤➠♥❣ ❣✐❛♥ ♠❡tr✐❝ ♥ã♥ ✳ ✳ ✳ ✷✵ ✷✳✶ ▼ét sè tÝ♥❤ ❝❤✃t t➠♣➠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✷✳✷ ❚Ý♥❤ ❝♦♠♣➝❝ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠❡tr✐❝ ♥ã♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ❑Õt ❧✉❐♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ❚➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✶ ❧ê✐ ♥ã✐ ➤➬✉ ❑❤➳✐ ♥✐Ư♠ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ ➤➢ỵ❝ ❍♦❛♥❣ ▲♦♥❣ ✲ ●✉❛♥❣ ✈➭ ❩❤❛♥❣ ❳✐❛♥ ➤➢❛ r❛ ♥➝♠ ✷✵✵✼ ❜➺♥❣ ❝➳❝❤ t❤❛② t❐♣ sè t❤ù❝ tr♦♥❣ ➤Þ♥❤ ♥❣❤Ü❛ ♠➟tr✐❝ ❜ë✐ ♠ét ♥ã♥ ➤Þ♥❤ ❤➢í♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ➤Þ♥❤ ❝❤✉➮♥✳ ❚➳❝ ❣✐➯ ❝ị♥❣ ➤➲ ①➞② ❞ù♥❣ ❝➳❝ ❦❤➳✐ ♥✐Ư♠ ✈Ị ❤é✐ tơ ❝đ❛ ❞➲②✱ tÝ♥❤ ➤➬② ➤đ ❝đ❛ ❦❤➠♥❣ ❣✐❛♥✱ ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ➤è✐ ✈í✐ ➳♥❤ ①➵ ❝♦✳✳✳ ✈➭ t❤✉ ợ ữ ết q s s tr ♥➭②✳ ◆❣➢ê✐ t❛ ➤➲ t❤✃② ➤➢ỵ❝ ♠ét sè ø♥❣ ❞ơ♥❣ ❝đ❛ ❧í♣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ tr♦♥❣ ❣✐➯✐ tÝ❝❤ ♣❤✐ t✉②Õ♥✱ tè✐ ➢✉ ✈❡❝t➡✳✳✳ ❍✐Ư♥ ♥❛② ♥❣❤✐➟♥ ❝ø✉ ❝✃✉ tró❝ ❝đ❛ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ ➤❛♥❣ t❤✉ ❤ót sù q✉❛♥ t➞♠ ❝ñ❛ ♠ét sè ♥❤➭ t♦➳♥ ❤ä❝ tr♦♥❣ ✈➭ ♥❣♦➭✐ ♥➢í❝✳ ❈➳❝ ✈✃♥ ➤Ị ✈Ị t➠♣➠ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ ➤➲ ➤➢ỵ❝ ♥❣❤✐➟♥ ❝ø✉ ë ♠ø❝ ➤é ❦❤ë✐ ➤➬✉✳ ❈➳❝ ✈✃♥ ➤Ò ✈Ò t➠♣➠ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠❡tr✐❝ ♥ã♥ ➤➢ỵ❝ ❙❤✳ ❘❡③❛♣♦✉r✱ ▼✳ ❉❡r❛❢s❤♣♦✉r✱ ❘✳ ❍❛♠❧❜❛r❛♥✐ ✈➭ ❝➳❝ ♥❤➭ t♦➳♥ ❤ä❝ ❦❤➳❝ ♥❣❤✐➟♥ ❝ø✉✱ ♠ë ré♥❣ t❤❡♦ ♥❤✐Ị✉ ❤➢í♥❣ ❦❤➳❝ ♥❤❛✉ ✈➭ ❜➢í❝ ➤➬✉ t❤✉ ➤➢ỵ❝ ♥❤✐Ị✉ t❤➭♥❤ tù✉✳ ❚r➟♥ ❝➡ së ❝➳❝ ❜➭✐ ❜➳♦ ✧❈♦♥❡ ♠❡tr✐❝ s♣❛❝❡s ❛♥❞ ❢✐①❡❞ ♣♦✐♥t t❤❡♦✲ r❡♠s ♦❢ ❝♦♥tr❛❝t✐✈❡ ♠❛♣♣✐♥❣s✧ ✈➭ ✧❆ r❡✈✐❡✇ ♦♥ t♦♣♦❧♦❣✐❝❛❧ ♣r♦♣❡rt✐❡s ♦❢ ❝♦♥❡ ♠❡tr✐❝ s♣❛❝❡✧✱ ❞➢í✐ sù ❤➢í♥❣ ❞➱♥ ❝đ❛ ◆●➛❚✳P●❙✳❚❙✳ ❚r➬♥ ❱➝♥ ➣♥✱ ❝❤ó♥❣ t➠✐ ➤➲ t✐Õ♣ ❝❐♥ ➤Ị t➭✐ ♥❣❤✐➟♥ ❝ø✉ ✧➜Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ➳♥❤ ①➵ ❝♦ rót ✈➭ ❝➳❝ tÝ♥❤ ❝❤✃t t➠♣➠ tr➟♥ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥✧✳ ▼ơ❝ ➤Ý❝❤ ❝đ❛ ❧✉❐♥ ✈➝♥ ♥➭② ❧➭ tr×♥❤ ❜➭② ❝➳❝ tÝ♥❤ ❝❤✃t ❝➳❝ tÝ♥❤ ❝❤✃t ❝đ❛ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥✱ ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ➳♥❤ ①➵ ❝♦ rót✱ ❝➳❝ tÝ♥❤ ❝❤✃t t➠♣➠ tr➟♥ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥✱ tÝ♥❤ ❝♦♠♣➝❝ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥✳ ◆❣♦➭✐ ♣❤➬♥ ▼ơ❝ ❧ơ❝✱ ❑Õt ❧✉❐♥ ✈➭ ❚➭✐ ❧✐Ư✉ t ộ ợ trì tr ❤❛✐ ❝❤➢➡♥❣✳ ✷ ❈❤➢➡♥❣ ✶✳ ➜Þ♥❤ ❧ý ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ➳♥❤ ①➵ ❝♦rót tr➟♥ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥✳ ❈❤➢➡♥❣ ✷✳ ❈➳❝ tÝ♥❤ ❝❤✃t t➠♣➠ ❝ñ❛ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥✳ ▲✉❐♥ ✈➝♥ ➤➢ỵ❝ t❤ù❝ ❤✐Ư♥ t➵✐ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤ ❞➢í✐ sù ❤➢í♥❣ ❞➱♥ t❐♥ t×♥❤ ✈➭ ♥❣❤✐➟♠ ❦❤➽❝ ❝ñ❛ ◆●➛❚✳P●❙✳❚❙✳ ❚r➬♥ ❱➝♥ ➣♥✳ ❚➳❝ ❣✐➯ ①✐♥ ❜➭② tá ò ết s s ủ ì ế ❞Þ♣ ♥➭②✱ t➳❝ ❣✐➯ ①✐♥ ❝❤➞♥ t❤➭♥❤ ❝➯♠ ➡♥ ❇❛♥ ❈❤đ ♥❤✐Ư♠ ❑❤♦❛ ❙❛✉ ➤➵✐ ❤ä❝✱ ❇❛♥ ❈❤đ ♥❤✐Ư♠ ❑❤♦❛ ❚♦➳♥✳ ❚➳❝ ❣✐➯ ①✐♥ ➤➢ỵ❝ ❝➯♠ ➡♥ ❝➳❝ t❤➬②✱ ❝➠ ❣✐➳♦ tr♦♥❣ ❑❤♦❛ ❚♦➳♥✱ ➤➷❝ ❜✐Ưt tr♦♥❣ ❚ỉ ●✐➯✐ tÝ❝❤ ➤➲ ♥❤✐Ưt t×♥❤ ❣✐➯♥❣ ❞➵② ✈➭ ❣✐ó♣ ➤ì t➳❝ ❣✐➯ tr♦♥❣ s✉èt t❤ê✐ ❣✐❛♥ ❤ä❝ t❐♣✳ ❈✉è✐ ❝ï♥❣✱ ①✐♥ ❝➯♠ ➡♥ ❣✐❛ ➤×♥❤✱ ➤å♥❣ ♥❣❤✐Ư♣✱ ❜➵♥ ❜❒✱ ❝➳❝ ❜➵♥ tr♦♥❣ ❧í♣ ❈❛♦ ❤ä❝ ✶✼ ✲ ●✐➯✐ tÝ❝❤ ➤➲ ❝é♥❣ t➳❝✱ ❣✐ó♣ ➤ì t➳❝ ❣✐➯ tr♦♥❣ s✉èt q✉➳ tr×♥❤ ❤ä❝ t❐♣ ✈➭ ♥❣❤✐➟♥ ❝ø✉✳ ▼➷❝ ❞ï ➤➲ ❝ã ♥❤✐Ò✉ ❝è ❣➽♥❣ ♥❤➢♥❣ ❧✉❐♥ ✈➝♥ ❦❤➠♥❣ tr➳♥❤ ❦❤á✐ ♥❤÷♥❣ ❤➵♥ ❝❤Õ✱ t❤✐Õ✉ sót ú t rt ợ ữ ý ế ➤ã♥❣ ❣ã♣ ❝ñ❛ ❝➳❝ t❤➬②✱ ❝➠ ❣✐➳♦ ✈➭ ❝➳❝ ❜➵♥ ➤Ĩ ❧✉❐♥ ✈➝♥ ➤➢ỵ❝ ❤♦➭♥ t❤✐Ư♥ ❤➡♥✳ ❱✐♥❤✱ t❤➳♥❣ ✶✷ ♥➝♠ ✷✵✶✶ ❚➳❝ ❣✐➯ ✸ ❝❤➢➡♥❣ ✶ ➜Þ♥❤ ❧ý ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ➳♥❤ ①➵ ❝♦ rót tr➟♥ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ ✶✳✶✳ ❑❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ ✶✳✶✳✶✳ ➜Þ♥❤ ♥❣❤Ü❛✳ K (K = R, C)✳ ❚❐♣ ❝♦♥ ❈❤♦ E ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ➤Þ♥❤ ❝❤✉➮♥ tr➟♥ tr➢ê♥❣ P E ❝đ❛ ➤➢ỵ❝ ❣ä✐ ❧➭ ♠ét ♥ã♥ tr♦♥❣ E ♥Õ✉ t❤á❛ ♠➲♥ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ s❛✉✿ ✐✮ P ❧➭ ➤ã♥❣✱ ❦❤➳❝ rỗ ọ ế x, y tộ P x t❤✉é❝ P ✶✳✶✳✷✳ ❱Ý ❞ô✳ ✈➭ P = {0}❀ a, b t❤✉é❝ R a, b ≥ t❤× ax + by t❤✉é❝ P ❀ ✈➭ −x t❤✉é❝ P ✶✮ ❳Ðt ❦❤➠♥❣ ❣✐❛♥ t❤× x = 0✳ R ✈í✐ ❝❤✉➮♥ t❤➠♥❣ t❤➢ê♥❣✳ ❑❤✐ ➤ã P = {x ∈ R : x ≥ 0} ❧➭ ♠ét ♥ã♥ tr♦♥❣ R✳ ✷✮ ❚r♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ➤Þ♥❤ ❝❤✉➮♥ R2 ✱ t❐♣ P = {(x, y) ∈ R2 : x, y ≥ 0} ❧➭ ♠ét ♥ã♥✳ ❈❤♦ P ❧➭ ♠ét ♥ã♥ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ➤Þ♥❤ ❝❤✉➮♥ q✉❛♥ ❤Ư t❤ø tù ✧≤✧ ①➳❝ ➤Þ♥❤ ❜ë✐ t❤✉é❝ ♥Õ✉ P✳ ❈❤ó♥❣ t❛ q✉② ➢í❝ y − x t❤✉é❝ ✐♥tP ✈í✐ ✶✳✶✳✸✳ ➜Þ♥❤ ♥❣❤Ü❛✳ x s❛♦ ❝❤♦ ✈í✐ ♠ä✐ ✳ ❙è ❞➢➡♥❣ K ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ tr➟♥ ➤➢ỵ❝ ❣ä✐ ❧➭ ❤➺♥❣ sè ❝❤✉➮♥ t➽❝ ❝đ❛ P✳ t❤✉é❝ ✹ x, y ♥❤á ♥❤✃t t❤á❛ ✷✮ P ➤➢ỵ❝ ❣ä✐ ❧➭ ❝❤Ý♥❤ q✉② ♥Õ✉ ♠ä✐ ❞➲② t➝♥❣ ✈➭ ❜Þ ❝❤➷♥ tr➟♥ tr♦♥❣ E ➤Ị✉ ❤é✐ tơ ✭♠ét ❝➳❝❤ t➢➡♥❣ ➤➢➡♥❣ ❧➭ ♠ä✐ ❞➲② ❣✐➯♠ ✈➭ ❜Þ ❝❤➷♥ ❞➢í✐ tr♦♥❣ E ➤Ị✉ ❤é✐ tơ✮✳ ✸✮ E P ➤➢ỵ❝ ❣ä✐ ❧➭ ♥ã♥ ♠✐♥✐❤❡❞r❛❧ ♥Õ✉ sup{x, y} tå♥ t➵✐ ✈í✐ ♠ä✐ x, y ∈ ✈➭ ❧➭ ♠✐♥✐❤❡❞r❛❧ ♠➵♥❤ ♥Õ✉ ♠ä✐ t❐♣ ❝♦♥ ❜Þ ❝❤➷♥ tr➟♥ ❝đ❛ E ➤Ị✉ ❝ã s✉♣♣r❡♠✉♠✳ ➜Þ♥❤ ❧ý s❛✉ ♥➟✉ ❧➟♥ ♠è✐ q✉❛♥ ❤Ư ❣✐÷❛ ♥ã♥ ❝❤✉➮♥ t➽❝ ✈➭ ♥ã♥ ❝❤Ý♥❤ q✉②✳ ✭❬✹❪✮✳ ▼ä✐ ♥ã♥ ❝❤Ý♥❤ q✉② ❧➭ ❝❤✉➮♥ t➽❝✳ ✶✳✶✳✹✳ ➜Þ♥❤ ❧ý ❈❤ø♥❣ ♠✐♥❤✳ ●✐➯ sư ó ỗ E P n t❛ ❝❤ä♥ ➤➢ỵ❝ tn , sn n2 tn < sn P ó í q ỗ tộ n ≥ 1✱ ➤➷t yn = P tn tn ❦❤➠♥❣ ❝❤✉➮♥ t➽❝✳ s❛♦ ❝❤♦ tn ✈➭ − sn t❤✉é❝ sn tn ✳ ❚❛ ❝ã xn = xn , yn , yn − xn ∈ E, yn = ✈➭ ∞ n ≤ xn ✱ ✈í✐ ♠ä✐ n ≥ 1✳ ì ỗ n=1 yn n2 ộ tụ tr n=1 ❇➞② ❣✐ê✱ ✈× E ✳ ❉♦ P |yn | n2 ➤ã♥❣ s✉② r❛ tå♥ t➵✐ ∞ = n=1 y∈P n2 ộ tụ ỗ s n=1 yn n2 = y✳ xn ≤ yn ✈➭ ❝➳❝❤ ①➳❝ ➤Þ♥❤ ủ ỗ tr s r x1 x1 + x2 x2 x3 ≤ x + + ≤ · · · ≤ y 22 22 32 ∞ xn lim xnn2 = 0✳ ❉♦ ➤ã✱ t❛ n2 ❤é✐ tô✳ ❙✉② r❛ n→∞ n=1 ♥❤❐♥ ➤➢ỵ❝ sù ♠➞✉ t❤✉➱♥ ✈í✐ n ≤ |xn ✳ ❱❐② ♠ä✐ ♥ã♥ ❝❤Ý♥❤ q✉② ì P í q ỗ t í ụ s t ệ ề ợ ủ ị ❧ý tr➟♥ ♥ã✐ ❝❤✉♥❣ ❦❤➠♥❣ ➤ó♥❣✳ ✶✳✶✳✺✳ ❱Ý ❞ơ✳ f ≥ 0}✳ ❑❤✐ ➤ã ❳Ðt P E = C[0,1] ✈í✐ ❝❤✉➮♥ ✧♠❛①✧ ✈➭ ♥ã♥ ❧➭ ♥ã♥ ❝❤✉➮♥ t➽❝✳ ❚❤❐t ✈❐②✱ ❣✐➯ sö ✺ P = {f ∈ E : f, g t❤✉é❝ E ✈➭ ≤ f ≤ g ✳ ❑❤✐ ➤ã ≤ f (x) ≤ g(x)✱ ✈í✐ ♠ä✐ x t❤✉é❝ [0, 1]✱ s✉② r❛ f = max |f (x)| = max f (x) ≤ max g(x) = max |g(x)| = g x∈[0,1] ❱❐② P x∈[0,1] x∈[0,1] x∈[0,1] ❧➭ ♥ã♥ ❝❤✉➮♥ t➽❝✳ ❳Ðt ❞➲② {fn } tr♦♥❣ E ①➳❝ ➤Þ♥❤ ♥❤➢ s❛✉✿ fn (x) = xn ✈í✐ ♠ä✐ x t❤✉é❝ (0, 1)✳ ❑❤✐ ➤ã ≤ · · · ≤ xn ≤ · · · ≤ x2 ≤ x, ✈í✐ ♠ä✐ x ∈ [0, 1]✳ ❙✉② r❛ ❞➲② tơ tr♦♥❣ {fn } ❣✐➯♠ ✈➭ ❜Þ ❝❤➷♥ ❞➢í✐✳ ❚✉② ♥❤✐➟♥ ❞➲② ♥➭② ❦❤➠♥❣ ❤é✐ E ✳ ❱❐② ♥ã♥ P ❦❤➠♥❣ ❝❤Ý♥❤ q✉②✳ ❚r♦♥❣ ♣❤➬♥ t✐Õ♣ t❤❡♦✱ t❛ ❧✉➠♥ ①Ðt ♥ã♥ tr♦♥❣ ✐✮ ❧➭ ❦❤➠♥❣ ❣✐❛♥ ❇❛♥❛❝❤✱ P ❧➭ ♠ét E ✈í✐ ✐♥tP = ∅ ✈➭ ✧≤✧ ❧➭ q✉❛♥ ❤Ư t❤ø tù tr➟♥ E ①➳❝ ➤Þ♥❤ ❜ë✐ P ✳ ✶✳✶✳✻✳ ➜Þ♥❤ ♥❣❤Ü❛✳ X→E E ❈❤♦ X ❧➭ ♠ét t❐♣ ❦❤➳❝ rỗ d : X ì ợ ọ ♠ét ♠➟tr✐❝ ♥ã♥ ♥Õ✉ t❤á❛ ♠➲♥ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ s❛✉ ≤ d(x, y) ✈í✐ ♠ä✐ x, y t❤✉é❝ X; d(x, y) = ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ x = y ❀ ✐✐✮ d(x, y) = d(y, x) ✈í✐ ♠ä✐ x, y t❤✉é❝ X ❀ ✐✐✐✮ d(x, y) ≤ d(x, z) + d(z, y) ✈í✐ ♠ä✐ x, y, z ❑❤✐ ➤ã t❤✉é❝ X✳ (X, d) ➤➢ỵ❝ ❣ä✐ ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠❡tr✐❝ ♥ã♥✳ ❑❤➠♥❣ ❣✐❛♥ ♠❡tr✐❝ ♥ã♥ ❧➭ sù tæ♥❣ q✉➳t ❝đ❛ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝✳ ❱Ý ❞ơ s❛✉ ❝❤ø♥❣ tá ❧í♣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ ❧➭ sù ♠ë ré♥❣ t❤ù❝ sù ❝đ❛ ❧í♣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝✳ ✶✳✶✳✼✳ ❱Ý ❞ơ✳ ❈❤♦ E = R2 ✈➭ P = {(x, y) ∈ E : x, y ≥ 0} ❳Ðt X = R ✈➭ ➳♥❤ d : X ì XE ị d(x, y) = (|x − y|, α|x − y|) ✈í✐ ♠ä✐ x, y ✻ t❤✉é❝ X, tr♦♥❣ ➤ã α ❧➭ sè t❤ù❝ ❞➢➡♥❣ ❝❤♦ tr➢í ❑❤✐ ➤ã✱ ❞Ơ ❞➭♥❣ ❦✐Ĩ♠ tr❛ ➤➢ỵ❝ (X, d) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥✳ ❙❛✉ ➤➞② ❝❤ó♥❣ t❛ tr×♥❤ ❜➭② ❝➳❝ ✈✃♥ ➤Ị ✈Ị sù ❤é✐ tơ ❝đ❛ ❞➲② tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥✳ ✶✳✶✳✽✳ ➜Þ♥❤ ♥❣❤Ü❛✳ ❧➭ ♠ét ❞➲② tr♦♥❣ ♠ä✐ c∈E ✈➭ X ✈➭ ❈❤♦ (X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥✱ {xn } x ∈ X ✳ ❉➲② {xn } ➤➢ỵ❝ ❣ä✐ ❧➭ ❤é✐ tơ tí✐ x ♥Õ✉ ✈í✐ c✱ tå♥ t➵✐ sè tù ♥❤✐➟♥ N d(xn , x) ❑❤✐ ➤ã ❤♦➷❝ x c, s❛♦ ❝❤♦ ✈í✐ ♠ä✐ ➤➢ỵ❝ ❣ä✐ ❧➭ ❣✐í✐ ❤➵♥ ❝đ❛ ❞➲② n > N {xn } ✈➭ t❛ ❦ý ❤✐Ö✉ xn → x (n→∞)✳ ✭❬✸❪✮✳ ❈❤♦ ✶✳✶✳✾✳ ▼Ư♥❤ ➤Ị (X, d) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥✱ ♥ã♥ ❝❤✉➮♥ t➽❝ ✈í✐ ❤➺♥❣ sè ❝❤✉➮♥ t➽❝ ❑✳ ●✐➯ sư ❳✳ ❑❤✐ ➤ã c ∈ E P ❧➭ {xn } ❧➭ ♠ét ❞➲② tr♦♥❣ {xn } ❤é✐ tơ tí✐ ① ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ d(xn , x) → (n→∞)✳ ❈❤ø♥❣ ♠✐♥❤✳ ●✐➯ sö s❛♦ ❝❤♦ c tå♥ t➵✐ sè tù ♥❤✐➟♥ N {xn } ❤é✐ tô ➤Õ♥ x✳ ✈➭ K c < ε✳ K ❱í✐ ♠ä✐ sè t❤ù❝ ❑❤✐ ➤ã tõ d(xn , x) s❛♦ ❝❤♦ ❝❤✉➮♥ t➽❝ ✈í✐ ❤➺♥❣ sè ❱❐② lim xn = x n→∞ c✱ xn → x (n→∞) ✈í✐ ♠ä✐ n > N✳ d(xn , x) ≤ K c < ε✱ ♥➟♥ ε > ❝❤ä♥ s✉② r❛ ❱× ♥ã♥ P n > N✳ ✈í✐ ♠ä✐ d(xn , x) → 0(n→∞)✳ ◆❣➢ỵ❝ ❧➵✐✱ ❣✐➯ sư tå♥ t➵✐ δ > s❛♦ ♠ë✮✳ ❱í✐ sè d(xn , x) → (n→∞)✳ ❚❛ ❝ã✱ ✈í✐ ♠ä✐ c ∈ E, ❝❤♦ x N ➜✐Ị✉ ♥➭② ♥❣❤Ü❛ ❧➭ ✼ ✭❞♦ s❛♦ ❝❤♦ ✈í✐ ♠ä✐ n > N ✳ ❱× t❤Õ {xn } ❤é✐ tơ tí✐ x P ✐♥t c d(xn , x) c ✈í✐ ♠ä✐ ✶✳✶✳✶✵✳ ▼Ư♥❤ ➤Ị ✭❬✸❪✮✳ ❈❤♦ (X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥✱ P ❧➭ ♠ét ♥ã♥ ❝❤✉➮♥ t➽❝ ✈í✐ ❤➺♥❣ sè ❝❤✉➮♥ t➽❝ ❑✳ ●✐➯ sư tr♦♥❣ X ✳ ◆Õ✉ {xn } ❤é✐ tơ tí✐ ❝➯ x ✈➭ y ❈❤ø♥❣ ♠✐♥❤✳ ▲✃② ❜✃t ❦ú s❛♦ ❝❤♦ d(xn , x) t❤× c ∈ E, c ✈➭ d(xn , y) x = y✳ c✳ ❑❤✐ ➤ã tå♥ t➵✐ sè tù ♥❤✐➟♥ N c, ✈í✐ ♠ä✐ n > N ✳ ❚õ ➤ã t❛ ❝ã d(x, y) ≤ d(xn , x) + d(xn , y) ❉♦ ➤ã t❛ ❝ã d(x, y) ≤ 2K c {xn } ❧➭ ♠ét ❞➲② ✳ ❱× 2c c tï② ý t❛ ❝ã d(x, y) = 0✳ ❉♦ ➤ã t❛ ❝ã x = y ▼Ư♥❤ ➤Ị s❛✉ ♥ã✐ ❧➟♥ tÝ♥❤ ❧✐➟♥ tơ❝ ❝đ❛ ➳♥❤ ①➵ ♠➟tr✐❝ ♥ã♥✳ ✶✳✶✳✶✶✳ ▼Ư♥❤ ➤Ị ✭❬✸❪✮✳ ❈❤♦ (X, d) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥✱ P ❧➭ ♥ã♥ ❝❤✉➮♥ t➽❝ ✈í✐ ❤➺♥❣ sè ❝❤✉➮♥ t➽❝ ❑ ✈➭ ◆Õ✉ xn → x (n→∞) ✈➭ yn → y (n→∞) t❤× d(xn , yn ) d(x, y) (n) ứ ỗ {xn }, {yn } ❧➭ ❝➳❝ ❞➲② tr♦♥❣ X ✳ > 0✱ ❝❤ä♥ c ∈ E s❛♦ ❝❤♦ xn → x ✈➭ yn → y ✱ tå♥ t➵✐ sè tù ♥❤✐➟♥ N d(xn , x) c ✈➭ d(yn , y) c, c ✈➭ c < ε 4K+2 ✳ s❛♦ ❝❤♦ ✈í✐ ♠ä✐ n > N ❉♦ ➤ã t❛ ❝ã d(xn , yn ) ≤ d(xn , x) + d(x, y) + d(y, yn ) ≤ d(x, y) + 2c, ✈í✐ ♠ä✐ n>N ✈➭ d(x, y) ≤ d(xn , x) + d(xn , yn ) + d(y, yn ) ≤ d(xn , yn ) + 2c, ❙✉② r❛ ✈í✐ ♠ä✐ n > N ≤ d(x, y) + 2c − d(xn , yn ) ≤ 4c ❚õ tÝ♥❤ ❝❤✉➮♥ t➽❝ ❝đ❛ ♥ã♥ ✈➭ ❜✃t ➤➻♥❣ t❤ø❝ tr➟♥ t❛ ♥❤❐♥ ➤➢ỵ❝ d(xn , yn ) − d(x, y) ≤ d(x, y) + 2c − d(xn , yn ) + 2c ≤ (K + 2) c < ε, ✽ ✈í✐ ♠ä✐ n > N ✳ ❇✃t ➤➻♥❣ t❤ø❝ tr➟♥ ❝❤ø♥❣ tá d(xn , yn )d(x, y) t trì ị ♥❣❤Ü❛ ❦❤➳✐ ♥✐Ö♠ ❞➲② ❈❛✉❝❤② tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥✳ ✶✳✶✳✶✷✳ ➜Þ♥❤ ♥❣❤Ü❛✳ tr♦♥❣ X ❈❤♦ (X, d) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥✳ ❉➲② {xn } ➤➢ỵ❝ ❣ä✐ ❧➭ ❞➲② ❈❛✉❝❤② ♥Õ✉ ✈í✐ ♠ä✐ c ∈ E, c✱ tå♥ t➵✐ sè tù ♥❤✐➟♥ ◆ s❛♦ ❝❤♦ d(xm , xn ) ✶✳✶✳✶✸✳ ▼Ư♥❤ ➤Ị ♥ã♥ ❝❤✉➮♥ t➽❝ ✈➭ ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ ✭❬✸❪✮✳ ❈❤♦ ✈í✐ ♠ä✐ (X, d) m, n > N ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥✱ P ❧➭ {xn } ❧➭ ♠ét ❞➲② tr♦♥❣ ❳✳ ❑❤✐ ➤ã {xn } ❧➭ ❞➲② ❈❛✉❝❤② d(xn , xm ) → (m, n→∞)✳ ❈❤ø♥❣ ♠✐♥❤✳ ●✐➯ sư ❝❤✉➮♥ t➽❝ ❝đ❛ c, {xn } ❧➭ ❞➲② ❈❛✉❝❤② tr♦♥❣ X ✳ P ✳ ❱í✐ ♠ä✐ ε > 0✱ ❝❤ä♥ c ∈ E s❛♦ ❝❤♦ ●ä✐ K ❧➭ ❤➺♥❣ sè c ✈➭ K c < ε✳ ❑❤✐ ➤ã tõ {xn } ❧➭ ❞➲② ❈❛✉❝❤②✱ tå♥ t➵✐ sè tù ♥❤✐➟♥ N s❛♦ ❝❤♦ d(xn , xm ) ✈í✐ ♠ä✐ n, m > N ✳ ❱× ♥ã♥ P ❧➭ ❝❤✉➮♥ t➽❝ ✈í✐ ❤➺♥❣ sè d(xn , xm ) ≤ K c < ε, ❱❐② ✈í✐ ♠ä✐ K c✱ ♥➟♥ m, n > N d(xn , xm ) → tr♦♥❣ E ✳ ◆❣➢ỵ❝ ❧➵✐✱ ❣✐➯ sư d(xn , xm ) → (m, n→∞) ❚❛ ❝ã ✈í✐ ♠ä✐ t❤✉é❝ c ∈ E, c✱ tå♥ t➵✐ δ > s❛♦ ❝❤♦ ♥Õ✉ x < δ t❤× c−x P ✳ ❱í✐ δ > ①➳❝ ➤Þ♥❤ ♥❤➢ tr➟♥✱ ❞♦ d(xn , xm ) → ❦❤✐ m, n → ∞ ✐♥t ♥➟♥ tå♥ t➵✐ N s❛♦ ❝❤♦ d(xn , xm ) < δ, ✈í✐ ♠ä✐ ✾ n, m > N ❙✉② r❛ c − d(xn , xm ) t❤✉é❝ P✳ ✐♥t d(xn , xm ) ❚❛ ♥❤❐♥ ➤➢ỵ❝ c ✈í✐ ♠ä✐ n, m > N ✱ tø❝ ❧➭ {xn } ❧➭ ❞➲② ❈❛✉❝❤②✳ ✶✳✶✳✶✹✳ ▼Ư♥❤ ➤Ị {xn } ✭❬✸❪✮✳ ❈❤♦ ❧➭ ♠ét ❞➲② tr♦♥❣ ❳✳ ◆Õ✉ (X, d) {xn } ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥✱ ❤é✐ tơ tr♦♥❣ (X, d) t❤× ♥ã ❧➭ ❞➲② ❈❛✉❝❤②✳ {xn } ❈❤ø♥❣ ♠✐♥❤✳ ●✐➯ sö t❤✉é❝ E ✱ tå♥ t➵✐ sè tù ♥❤✐➟♥ N ❤é✐ tơ tí✐ m, n > N c , ✈í✐ ♠ä✐ c n > N t❛ ❝ã d(xm , xn ) ≤ d(xn , x) + d(xm , x) ❱× t❤Õ ❑❤✐ ➤ã✱ ✈í✐ ♠ä✐ s❛♦ ❝❤♦ d(xn , x) ❱× ✈❐②✱ ✈í✐ ♠ä✐ x ∈ X✳ c c + = c 2 {xn } ❧➭ ❞➲② ❈❛✉❝❤②✳ ✶✳✶✳✶✺✳ ➜Þ♥❤ ♥❣❤Ü❛✳ ❑❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ ➤đ ♥Õ✉ ♠ä✐ ❞➲② ❈❛✉❝❤② ➤Ị✉ ❤é✐ tơ tr♦♥❣ ✶✳✶✳✶✻✳ ➜Þ♥❤ ♥❣❤Ü❛✳ ✈í✐ ❜✃t ❦ú ❞➲② ❈❤♦ (X, d) ➤➢ỵ❝ ❣ä✐ ❧➭ ➤➬② X✳ (X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥✳ ◆Õ✉ {xn } tr♦♥❣ X ✱ tå♥ t➵✐ ❞➲② ❝♦♥ {xni } ❝ñ❛ {xn } s❛♦ ❝❤♦ {xni } ❧➭ ❞➲② ❤é✐ tô tr♦♥❣ X tì X ợ ọ tr ó ❞➲②✳ ✶✳✷✳ ➜Þ♥❤ ❧ý ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ➳♥❤ ①➵ ❝♦ rót ❚r♦♥❣ ♠ơ❝ ♥➭②✱ ❝❤ó♥❣ t❛ tr×♥❤ ❜➭② ♠ét sè ➤Þ♥❤ ❧ý ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ➳♥❤ ①➵ ❝♦ rót✳ ✶✳✷✳✶✳ ➜Þ♥❤ ❧ý✳ ❈❤♦ (X, d) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ ➤➬② ➤ñ✱ P ❧➭ ♠ét ♥ã♥ ❝❤✉➮♥ t➽❝ ✈í✐ ❤➺♥❣ sè ❝❤✉➮♥ t➽❝ ❑✳ ●✐➯ sư ✶✵ T : X−→X ❧➭ ➳♥❤ B⊆X ❚❐♣ ❝♦♥ t❤✉é❝ ➤➢ỵ❝ ❣ä✐ ❧➭ t❐♣ ➤ã♥❣ ♥Õ✉ ♠ä✐ ➤✐Ĩ♠ ❣✐í✐ ❤➵♥ ❝đ❛ B ➤Ị✉ B✳ ✸✮ ❚❐♣ ❝♦♥ A⊆X s❛♦ ❝❤♦ BX ợ ọ ị ế tồ t c ✈➭ x0 ∈ X c ✈í✐ ♠ä✐ b ∈ B d(b, x0 ) ✺✮ ❚❐♣ ❝♦♥ A ➤Ò✉ ❧➭ A✳ ➤✐Ĩ♠ tr♦♥❣ ❝đ❛ ✹✮ ❚❐♣ ❝♦♥ ➤➢ỵ❝ ❣ä✐ ❧➭ t ế ỗ tử ủ B X ợ ❣ä✐ ❧➭ t❐♣ ❝♦♠♣➝❝ ♥Õ✉ ♠ä✐ ♣❤ñ ♠ë ❝ñ❛ B ➤Ị✉ ❝ã ♣❤đ ❝♦♥ ❤÷✉ ❤➵♥✳ ❚r♦♥❣ ♣❤➬♥ t✐Õ♣ t❤❡♦✱ t❛ ❧✉➠♥ ❣✐➯ t❤✐Õt ❧➭ ♠ét ♥ã♥ tr♦♥❣ E ❧➭ ❦❤➠♥❣ ❣✐❛♥ ✈❡❝t➡ t➠♣➠✱ P E ✈í✐ ✐♥tP = ∅ ✈➭ ✧≤✧ ❧➭ q✉❛♥ ❤Ö t❤ø tù tr➟♥ E ①➳❝ ➤Þ♥❤ P✳ ❜ë✐ ✷✳✶✳✹✳ ▼Ư♥❤ ➤Ị ✭❬✼❪✮✳ ❈❤♦ (X, d) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥✱ a, b, c ∈ X, A t rỗ ủ X ế ✐✐✮ a b ✈➭ b ✈➭ c t❤× a α ❧➭ ♠ét sè t❤ù❝ ❞➢➡♥❣✳ ❑❤✐ ➤ã c✳ α✐♥tP ⊆ ✐♥tP ✳ ✐✐✐✮ ◆Õ✉ a ≤ b ✈➭ b c tì a c ỗ > x ∈ ✐♥tP ✱ tå♥ t➵✐ < γ < s x < ỗ c1 d, c2 ✈➭ c2 ∈ P ✱ c1 ✈➭ 0 d e s❛♦ ❝❤♦ s❛♦ ❝❤♦ c2 ❝ã ♠ét ♣❤➬♥ tö e c2 ✳ ❈❤ø♥❣ ♠✐♥❤✳ ✐✮ ➜Ĩ ý r➺♥❣ t❐♣ ♠ë tr♦♥❣ ❝đ❛ tå♥ t ột tử d ỗ c1 , e c1 E ❧➭ ❦❤➠♥❣ ❣✐❛♥ ✈❡❝t➡ t➠♣➠ ✈➭ ✐♥tP ❧➭ ♠ét P ✳ ❱× ♣❤Ð♣ ❝é♥❣ ❧➭ ➤å♥❣ ♣❤➠✐ t❛ ❝ã x + ✐♥tP E ✳ ❉♦ ➤ã P + ✐♥tP = x + ✐♥tP ✐♥t x∈✐♥tP ✷✶ ❧➭ t❐♣ ❝♦♥ ♠ë ❝ò♥❣ ❧➭ t❐♣ ♠ë tr♦♥❣ P ⊆ ✐♥t E ✳ ▼➷t ❦❤➳❝✱ ✐♥tP + ✐♥tP ⊆ P ✳ ❚õ ➤ã s✉② r❛ ✐♥tP + P ✳ ❙ö ❞ơ♥❣ ❦Õt q✉➯ ♥➭② ✈➭♦ ❝❤ø♥❣ ♠✐♥❤ ❜ỉ ➤Ị t❛ ❝ã a c s✉② r❛ b − a ∈ ✐♥tP b b ✈➭ ✐♥t ✈➭ c − b ∈ ✐♥tP ✱ ❞♦ ➤ã c − a = (c − b) + (b − a) ∈ ✐♥tP + ✐♥tP ⊆ ✐♥tP ❤❛② a c✳ E ✐✐✮ ❱× ❧➭ ❦❤➠♥❣ ❣✐❛♥ ✈❡❝t➡ t➠♣➠ ♥➟♥ ♣❤Ð♣ ♥❤➞♥ ✈➠ ❤➢í♥❣ ❧➭ ➤å♥❣ ♣❤➠✐✱ ❞♦ ➤ã α✐♥tP ❧➭ t❐♣ ♠ë tr♦♥❣ E✳ ❑Õt ❤ỵ♣ ✈í✐ α✐♥tP ⊆ P t❛ ❝ã α✐♥tP ⊆ ✐♥tP ✳ ✐✐✐✮ ❱× x + ✐♥tP ❧➭ t❐♣ ♠ë ♥➟♥ P +P = P + P ⊆ ✐♥tP ✳ ❚õ a ≤ b ✈➭ b ❞♦ ➤ã c ✐♥t ➜✐Ò✉ ♥➭② ❦Ð♦ t❤❡♦ x + ✐♥tP ✐♥t x∈P t❛ ❝ã b − a ∈P ❝ò♥❣ ❧➭ t❐♣ ♠ë✱ ✈➭ c − b ∈ ✐♥tP ✳ c − a = c − b + b − a ∈ ✐♥tP + P ⊆ ✐♥tP ✳ δ > ✈➭ x ∈ ✐♥tP ✱ ❝❤ä♥ ♠ét sè tù ♥❤✐➟♥ n > s❛♦ ❝❤♦ δ δ δ < 1✳ ➜➷t γ = t❛ ❝ã < γ < ✈➭ γx = < 1✳ ❱❐② ợ n x n x n ỗ ❝❤ø♥❣ ♠✐♥❤✳ ✈✮ ❱× δ > tr♦♥❣ c1 s❛♦ ❝❤♦ E ♥➟♥ c1 ∈ c1 + B(δ, 0) ⊆ ✈í✐ ❜➳♥ ❦Ý♥❤ δ ✳ ❚õ c2 ∈ mB(δ, 0)✳ P✳ ❱× t❤Õ t❛ ❝ã ▼➭ ✐♥t P✱ ✐♥t P ✐♥t ✈í✐ ❧➭ t❐♣ ♠ë ♥➟♥ t❛ ❝❤ä♥ ➤➢ỵ❝ B(δ, 0) ❧➭ ❧➞♥ ❝❐♥ ❝❤✉➮♥ ❝ñ❛ B(δ, 0) ❧➭ t❐♣ ❤ót✱ t❛ ❝ã t❤Ĩ ❧✃② m > s❛♦ ❝❤♦ −c2 ∈ mB(δ, 0) ✈➭ mc1 − c2 ∈ m✐♥tP ⊆ P ✐♥t ✭t❤❡♦ ✐✐✮✳ ➜➷t d = mc1 t❛ ❝ã c1 d ✈➭ mc1 − c2 ∈ P ⇒ c2 ✐♥t d = mc1 t❤á❛ ♠➲♥ tÝ♥❤ t ợ ò ỏ c2 ó t ❝ã e ✈✐✮ ❚r♦♥❣ ❝❤ø♥❣ ♠✐♥❤ ✈✮✱ ➤➷t e = m c1 − e = c1 − ❱❐② e c2 ✈➭ e mc1 = d✳ c2 ✈➭ 1 c2 = (mc1 − c2 ) ∈ ✐♥tP ⊆ ✐♥tP m m m c1 ✷✷ ❱❐② t❐♣ ❝♦♥ ❦❤➳❝ rỗ ủ X ó ọ ộ tô tr♦♥❣ ✐✐✮ ◆Õ✉ x∈X (X, d) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ ✈➭ A ❧➭ ✭❬✼❪✮✳ ❈❤♦ ✷✳✶✳✺✳ ▼Ư♥❤ ➤Ị X ❧➭ ❜Þ ❝❤➷♥✳ ❧➭ ➤✐Ĩ♠ ❣✐í✐ ❤➵♥ ❝đ❛ A t❤× tå♥ t➵✐ ❞➲② {xn }n≥1 tr♦♥❣ A s❛♦ ❝❤♦ xn → x ❦❤✐ n → ∞✳ ✐✈✮ ❚❐♣ F X ỗ e x X, N (x, e) ❧➭ t❐♣ ♠ë✳ ✐✐✐✮ ❱í✐ ♠ä✐ ❧➭ t❐♣ ♠ë ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ c ✈➭ x ∈ X✱ t❐♣ F c = X\F ❧➭ t❐♣ ➤ã♥❣✳ B(x, c) := {y ∈ X : d(x, y) ≤ c} ❧➭ t❐♣ ➤ã♥❣✳ ❈❤ø♥❣ ♠✐♥❤✳ ✐✮ ●✐➯ sö ❦❤✐ ➤ã ❝ã sè tù ♥❤✐➟♥ ➤Ò ✷✳✶✳✹ tå♥ t➵✐ N e1 {xn }n≥1 s❛♦ ❝❤♦ s❛♦ ❝❤♦ X ✈➭ c✱ c ✈í✐ ♠ä✐ n > N ✳ ❚❤❡♦ ▼Ư♥❤ e1 ✈í✐ i = 1, 2, , N ✳ ▲➵✐ t❤❡♦ e s❛♦ ❝❤♦ e1 e ✈➭ c e✳ ❚õ ➤ã t❛ e ✈í✐ ♠ä✐ n ≥ 1✳ ❱× t❤Õ ❞➲② {xn }n≥1 ❧➭ ❞➲② ❜Þ ❝❤➷♥✳ d(xn , x) ✐✐✮ ❱í✐ d(xn , x) d(xi , x) ▼Ư♥❤ ➤Ị ✷✳✶✳✹ ♣❤➬♥ ✭✈✮✱ tå♥ t➵✐ ❝ã ❧➭ ♠ét ❞➲② ❤é✐ tô tr♦♥❣ e ❝❤♦ tr➢í❝✱ ✈× x ∈ X c ✈➭ ❧➭ ➤✐Ĩ♠ ❣✐í✐ ❤➵♥ ❝đ❛ A ♥➟♥ e ∩ (A\{x}) = ∅✱ ọ n ì tế ỗ n 1✱ ❝❤ä♥ ➤➢ỵ❝ n e x ∈ N x, ∩ (A\{x}) n e ✱ ♠ä✐ n ≥ 1✳ ❚❤❡♦ ▼Ö♥❤ ➤Ò ✷✳✶✳✹✱ tå♥ t➵✐ sè tù ♥❤✐➟♥ ❚õ ➤ã d(xn , x) n N > s❛♦ ❝❤♦ e N c ó d(xn , x) c ỗ n ≥ N ✳ ❱❐② ❞➲② N x, {xn }n≥1 ❤é✐ tơ tí✐ x✳ y ∈ N (x, e) t❛ sÏ y ❧➭ ➤✐Ĩ♠ tr♦♥❣ ❝đ❛ N (x, e)✳ ❱× y ∈ N (x, e) ♥➟♥ d(x, y) e✳ ✐✐✐✮ ●✐➯ sö ❝❤ø♥❣ ♠✐♥❤ ➜➷t e c = e − d(x, y) ✈❐②✱ ♥Õ✉ ✈➭ x ∈ X t❛ ❝ã c✳ z ∈ N (y, c) t❤× d(z, y) d(x, y) + d(y, z) e✳ ❉♦ ➤ã ❝❤♦ tr➢í❝✳ ❱í✐ ❜✃t ❦ú ❚❛ sÏ ❝❤Ø r❛ N (y, c) ⊆ N (x, e)✳ ❚❤❐t c = e − d(x, y)✳ ❉♦ ➤ã t❛ ❝ã d(x, z) ≤ z ∈ N (x, e)✳ ✷✸ ◆❤➢ ✈❐② y ❧➭ ➤✐Ó♠ tr♦♥❣ ❝ñ❛ N (x, e)✳ ❚õ ➤ã s✉② r❛ N (x, e) ❧➭ t❐♣ ♠ë✳ ✐✈✮ ❍✐Ĩ♥ ♥❤✐➟♥✳ ✈✮ ●✐➯ sư z∈X ❧➭ ➤✐Ĩ♠ ❣✐í✐ ❤➵♥ ❝đ❛ N z, ✈í✐ ♠ä✐ c ∩ B(x, c) = ∅ n n ≥ 1✳ ◆Õ✉ y ∈ N z, t❤× B(x, c)✱ t❛ ❝ã y ∈ N z, c n ✈➭ c ∩ B(x, c) n y ∈ B(x, c)✳ ❙✉② r❛ d(x, y) ≤ c ✈➭ d(y, z) d(x, y) + d(y, z) c ✳ ❉♦ ➤ã n c c+ n ❱× t❤Õ c c+ n d(x, z) ❚õ c = t❛ ❝ã d(x, z) ≤ c✳ n lim n→∞ ❉♦ ➤ã z ∈ B(x, c)✳ ❱❐② B(x, c) ❧➭ t❐♣ ➤ã♥❣✳ ✷✳✶✳✻✳ ▼Ư♥❤ ➤Ị c, c ✈➭ ✭❬✼❪✮✳ ●✐➯ sö (X, d) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥✱ x, y ∈ X ✳ ❑❤✐ ➤ã N (x, c) ∪ N (x, c ) ⊆ N (x, c + c )✳ c+c ✳ ✐✐✮ N (x, c) ∩ N (x, c ) ⊆ N x, ✐✐✐✮ ◆Õ✉ N (x, c) ∩ N (y, c ) = ∅ t❤× y ∈ N (x, c + c )✳ ✐✮ ✐✈✮ N (x, c) ∪ (y, c ) ⊆ N (x, c + c + d(x, y))✳ ❈❤ø♥❣ ♠✐♥❤✳ ✐✮ ●✐➯ sö y ∈ N (x, c )✳ ▼➷t ❦❤➳❝ ❱í✐ c y ∈ N (x, c) ∪ N (x, c )✱ t❛ ❝ã y ∈ N (x, c) ❤♦➷❝ y ∈ N (x, c)✱ ❦❤✐ ➤ã d(x, y) ♥➟♥ c ♥➟♥ c − d(x, y) ∈ c ∈ ✐♥tP ✳ ❚õ ➤ã c + c − d(x, y) ∈ ✐♥tP + ✐♥tP ⊆ ✐♥tP ✷✹ P✳ ✐♥t ❱× t❤Õ d(x, y) c+c ❤❛② ❝ị♥❣ ❝❤ø♥❣ ♠✐♥❤ ➤➢ỵ❝ y ∈ N (x, c + c ) ❚➢➡♥❣ tù✱ ✈í✐ y ∈ N (x, c ) t❛ y ∈ N (x, c + c )✳ ❱❐② N (x, c) ∪ N (x, c ) ⊆ N (x, c + c ) ✐✐✮ ●✐➯ sö y ∈ N (x, c) ∩ N (x, c )✱ ❦❤✐ ➤ã d(x, y) c − d(x, y) ∈ ✐♥tP ✈➭ c ✈➭ d(x, y) c ♥➟♥ c − d(x, y) ∈ ✐♥tP ✳ ❚õ ➤ã c + c − 2d(x, y) ∈ ✐♥tP ✳ ❱× t❤Õ ❉♦ ➤ã d(x, y) c+c − d(x, y) ∈ ✐♥tP ⊆ ✐♥tP 2 c+c c+c ❤❛② y ∈ N x, ❱❐② 2 N (x, c) ∩ N (x, c ) ⊆ N x, ✐✐✐✮ ●✐➯ sö ❙✉② r❛ N (x, c) ∩ N (y, c ) = ∅✳ z ∈ N (x, c) ✈➭ z ∈ N (y, c )✳ t❤Õ t❛ ❝ã c − d(x, z) ∈ ✐♥tP ✈➭ c+c ❑❤✐ ➤ã t❛ ❝ã ❉♦ ➤ã z ∈ N (x, c) ∩ N (y, c )✳ d(x, z) c ✈➭ d(y, z) c✳ ❱× c − d(y, z) ∈ ✐♥tP ✳ ❚õ ➤ã c + c − d(x, z) − d(y, z) ∈ ✐♥tP ❑Õt ❤ỵ♣ ✈í✐ d(x, y) d(x, y) ≤ d(x, z) + d(y, z) t❛ ❝ã c + c − d(x, y) ∈ ✐♥tP ✳ ❉♦ ➤ã c + c ✳ ❱❐② y ∈ N (x, c + c ) ✐✈✮ ❚➢➡♥❣ tù ❝❤ø♥❣ ♠✐♥❤ ë ✐✮✱ t❛ ❝ã N (x, c) ⊆ N (x, c + c + d(x, y)) ❇➞② ❣✐ê t❛ ❝ß♥ ♣❤➯✐ ❝❤ø♥❣ ♠✐♥❤ N (y, c ) ⊆ N (x, c + c + d(x, y)) ▲✃② z ∈ N (y, c )✱ t❛ ❝ã d(y, z) d(x, z) c ♥➟♥ c − d(y, z) ∈ ✐♥tP ✳ ❱× d(x, y) + d(z, y) + c ✷✺ ♥➟♥ d(x, y) + d(z, y) + c − d(x, z) ∈ ✐♥tP ❚õ ➤ã s✉② r❛ c − d(y, z) + d(x, y) + d(z, y) + c − d(x, z) ∈ ✐♥tP, ❤❛② c + c + d(x, y) − d(x, z) ∈ ✐♥tP ❙✉② r❛ d(x, z) c + c + d(x, y) ❚õ ➤ã t❛ ❝ã z ∈ N (x, c + c + d(x, y)) ❱× t❤Õ N (y, c ) ⊆ N (x, c + c + d(x, y)) ❱❐② N (x, c) ∩ N (y, c ) ⊆ N (x, c + c + d(x, y)) ✷✳✶✳✼✳ ▼Ư♥❤ ➤Ị ✭❬✼❪✮✳ ●✐➯ sư (X, d) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ ✐✮ ❱í✐ ❤ä ❜✃t ❦ú {Gα }α∈I ❝➳❝ t❐♣ ❝♦♥ ♠ë ❝ñ❛ ✐✐✮ ❱í✐ ❤ä ❜✃t ❦ú {Fα }α∈I ❝➳❝ t❐♣ ❝♦♥ ➤ã♥❣ ❝ñ❛ X, Gα ❧➭ t❐♣ ♠ë✳ α∈I X, Fα ❧➭ t❐♣ ➤ã♥❣✳ α∈I n ✐✐✐✮ ❱í✐ ❤ä ❤÷✉ ❤➵♥ G1 , G2 , , Gn ✱ ❝➳❝ t❐♣ ♠ë✱ Gi ❧➭ t❐♣ ♠ë✳ Fi ❧➭ t❐♣ ➤ã♥❣✳ i=1 n ✐✈✮ ❱í✐ ❤ä ❤÷✉ ❤➵♥ F1 , F2 , , Fn ❝➳❝ t❐♣ ➤ã♥❣✱ i=1 ❈❤ø♥❣ ♠✐♥❤✳ ✐✮ ▲✃② ❜✃t ❦ú x∈ Gα ✱ ❦❤✐ ➤ã tå♥ t➵✐ α0 ∈ I s❛♦ ❝❤♦ α∈I x ∈ Gα0 ✳ ❉♦ Gα0 r❛ N (x, p) ⊆ ❧➭ t❐♣ ♠ë ♥➟♥ tå♥ t➵✐ p s❛♦ ❝❤♦ N (x, p) ⊆ Gα0 ✳ ❙✉② Gα ✳ ❉♦ ➤ã x ❧➭ ➤✐Ĩ♠ tr♦♥❣ ❝đ❛ α∈I Gα ✳ ❱❐② α∈I ♠ë✳ ✷✻ Gα ❧➭ t❐♣ α∈I Fα , α ∈ I ✐✐✮ ●✐➯ sö ❧➭ t❐♣ ➤ã♥❣✱ ❦❤✐ ➤ã (X\Fα ) = X\ ❚❤❡♦ ✐✮ t❛ ❝ã α∈I Fα X\Fα ❧➭ t❐♣ ♠ë✱ ♠ä✐ α ∈ I✳ Fα ❧➭ t❐♣ ❧➭ t❐♣ ♠ë✱ ✈× t❤Õ α∈I α∈I ➤ã♥❣✳ n t ỳ x Gi ỗ i = 1, 2, , n✱ ✈× Gi i=1 ❝ã n ci s❛♦ ❝❤♦ N (x, ci ) ⊆ Gi ✳ ➜➷t N = ❧➭ t❐♣ ♠ë ♥➟♥ n N (x, ci ) t❛ ❝ã N ⊂ i=1 n i=1 Gi ❧➭ t❐♣ ♠ë t❛ ❝❤Ø ❝➬♥ ❝❤ø♥❣ ♠✐♥❤ N t❤Õ✱ ➤Ó ❝❤ø♥❣ ♠✐♥❤ Gi ✱ ✈× ❧➭ t❐♣ ♠ë✳ i=1 n ci ●✐➯ sư y ∈ N ✱ t❤❡♦ ♣❤➬♥ ✐✐✮ ❝đ❛ ▼Ư♥❤ ➤Ị ✷✳✶✳✻ t❛ ❝ã y ∈ N x, ➤ã n ci i=1 n − d(x, y) ∈ ✐♥tP n ci ▲✃② z ∈ N x, i=1n − d(x, y ✈➭ r > s❛♦ ❝❤♦ n ci t∈E: i=1 n − d(x, y) − d(z, y) − t < r ⊆ P ❚❛ sÏ ❝❤Ø r❛ r➺♥❣ n {t ∈ E : ci − d(x, z) − t < r} ⊆ P i=1 ◆Õ✉ ci − d(x, z) − t < r ✈í✐ ♠ä✐ ≤ i ≤ n t❤× n ci t − d(x, y) − d(z, y) + d(x, z) + i=1 ❚õ − ci ∈ P n n ci d(x, y) + d(y, z) − d(x, z) ∈ P, t + ✷✼ i=1 n − ci ∈ P i=1 n ✱ ❞♦ t❛ ❝ã n ci n nt = t+ i=1 ❱× t❤Õ i=1 n − ci ∈ P t ∈ P ✳ ❚õ ➤ã s✉② r❛ n {t ∈ E : ci − d(x, z) − t < r} ⊆ P i=1 n ❉♦ ➤ã z∈ N (x, ci ) ✈➭ ✈× t❤Õ i=1 n ci N y, ➜✐Ị✉ ➤ã ❝❤Ø r❛ r➺♥❣ ✐✈✮ ●✐➯ sư ✈í✐ ♠ä✐ N i=1 n − d(x, y) ⊆ N ❧➭ t❐♣ ♠ë ✈➭ ❤♦➭♥ t❤➭♥❤ ✈✐Ö❝ ❝❤ø♥❣ ♠✐♥❤✳ F1 , F2 , , Fn ❧➭ ❝➳❝ t❐♣ ➤ã♥❣✳ ❑❤✐ ➤ã t❛ ❝ã X\Fi ❧➭ t❐♣ ♠ë i = 1, 2, , n✳ ❚❤❡♦ ❦Õt q✉➯ ❝ñ❛ ♣❤➬♥ ✐✐✐✮ t❛ ❝ã n n (X\Fi ) = X\ i=1 Fi i=1 n Fi ❧➭ t❐♣ ➤ã♥❣✳ ❧➭ t❐♣ ♠ë✱ tõ ➤ã s✉② r❛ i=1 ✷✳✶✳✽✳ ▼Ư♥❤ ➤Ị ✭❬✽❪✮✳ ❈❤♦ (X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥✳ ❑❤✐ ➤ã ❤ä B = {N (x, e) : x ∈ X, ❧➭ ♠ét t✐Ị♥ ❝➡ së ❝đ❛ t➠♣➠ tr➟♥ X ✳ ❚❛ ❦ý ❤✐Ö✉ t➠♣➠ ♥ã♥ ♥➭② ❧➭ τc ✳ ❈❤ø♥❣ ♠✐♥❤✳ ❚❛ ❝❤Ø ❝➬♥ ❝❤ø♥❣ ♠✐♥❤ r➺♥❣ ❤❛✐ ♣❤➬♥ tư ❝đ❛ e} N (x1 , c1 ) ✈➭ B ✈➭ y ∈ N (x1 , c1 ) ∩ N (x2 , c2 ) t❤× tå♥ t➵✐ N (y, e) ⊆ N (x1 , c1 ) ∩ N (x2 , c2 ) ✷✽ N (x2 , c2 ) ❧➭ e s❛♦ ❝❤♦ ❚❤❐t ✈❐②✱ ❝❤ä♥ e1 N (y, e2 ) ⊆ N (x2 , c2 )✳ ✈➭ e ✈➭ e2 s❛♦ ❝❤♦ ❚❤❡♦ ▼Ư♥❤ ➤Ị ✷✳✶✳✹ ❝ã N (y, e2 ) ⊆ N (x1 , c1 ) e s❛♦ ❝❤♦ e ✈➭ e1 e2 ✳ ❑❤✐ ➤ã N (y, e) ⊆ N (y, e1 ) ⊆ N (x1 , c1 ) ✈➭ N (y, e) ⊆ N (y, e2 ) ⊆ N (x2 , c2 ) ✈× t❤Õ N (y, e) = N (x1 , c1 ) ∩ N (x2 , c2 ) ✷✳✶✳✾✳ ➜Þ♥❤ ❧ý✳ ❈❤♦ ♣❤➬♥ tư ♣❤➞♥ ❜✐Ưt (X, d) ❧➭ tr ó ỗ x, y X ✱ ❝ã c s❛♦ ❝❤♦ N (x, c) ∩ N (y, c) = ∅ ❈❤ø♥❣ ♠✐♥❤✳ ●✐➯ sö ♥❣➢ỵ❝ ❧➵✐ ❝ã N (y, c) = ∅ ✈í✐ ♠ä✐ c✳ x, y ∈ X, x = y ❚❤❡♦ ▼Ư♥❤ ➤Ị ✷✳✶✳✻✱ s❛♦ ❝❤♦ N (x, c) ∩ y ∈ N (x, 2c) ✈í✐ ♠ä✐ c c ✈í✐ ♠ä✐ n ≥ 1✳ ❱× t❤Õ − d(x, y) ∈ P ✈í✐ ♠ä✐ n n n ≥ 1✳ ❉♦ ➤ã −d(x, y) ∈ P ✳ ➜✐Ò✉ ♥➭② s✉② r❛ x = y ✳ ➜➞② ❧➭ ♠➞✉ t❤✉➱♥✳ c✳ ❚õ ➤ã y ∈ N x, ❚õ ➤Þ♥❤ ❧ý tr➟♥ t❛ ❝ã ❤Ö q✉➯ s❛✉✳ ✷✳✶✳✶✵✳ ❍Ö q✉➯✳ ❈❤♦ τc (X, d) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥✳ ❑❤✐ ➤ã t➠♣➠ ♥ã✐ tr♦♥❣ ▼Ư♥❤ ➤Ị ✷✳✶✳✽ ❧➭ ♠ét t➠♣➠ ❍❛✉s❞♦r❢❢✳ ✷✳✷✳ ❚Ý♥❤ ❝♦♠♣➝❝ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ ❚r♦♥❣ ♠ô❝ ♥➭②✱ t❛ ❝❤Ø r❛ ❝➳❝ tÝ♥❤ ❝❤✃t ❝♦♠♣➝❝ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ t➢➡♥❣ tù ♥❤➢ ❝➳❝ tÝ♥❤ ❝❤✃t ❝♦♠♣➝❝ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝✳ ➜å♥❣ t❤ê✐ ❝❤ø♥❣ ♠✐♥❤ ♠ét sè ❦Õt q✉➯ ✈Ò tÝ♥❤ ❝♦♠♣➝❝ ❝ñ❛ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥✳ ë ➤➞②✱ ❝➳❝ ❦❤➳✐ ♥✐Ö♠ ❝♦♠♣➝❝ ✈➭ ❝♦♠♣➝❝ ❞➲② ợ ị ĩ t tự ệ ❝ã tr➟♥ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ t❤➠♥❣ t❤➢ê♥❣✳ ✷✾ ✷✳✷✳✶✳ ▼Ö♥❤ ề ế ỗ {A }I D X ✭❬✼❪✮✳ ❈❤♦ (X, d) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ ❝♦♠♣➝❝ ❧➭ ♠ét ♣❤đ ♠ë ❝đ❛ d(x, y) c ✈í✐ ♠ä✐ X t❤× tå♥ t➵✐ x, y ∈ D✱ tå♥ t➵✐ c s❛♦ ❝❤♦ ✈í✐ α0 ∈ I s❛♦ ❝❤♦ D ⊆ Aα0 ✳ ❈❤ø♥❣ ♠✐♥❤✳ ●✐➯ sö r➺♥❣ ♣❤➳t ể tr ú ó ỗ Ba a tå♥ t➵✐ Ba ⊆ X ❦❤➠♥❣ ➤➢ỵ❝ ❝❤ø❛ tr♦♥❣ ❝❤ä♥ Bn ⊆ X ❝❤ø❛ tr♦♥❣ s❛♦ ❝❤♦ Aα Aα ♥➭♦✳ ❱í✐ ♥➭♦✳ ❈❤ä♥ ✈➭ ❤é✐ tơ tí✐ ➤✐Ĩ♠ t❐♣ ♠ë ♥➟♥ ❝ã K s❛♦ ❝❤♦ f f e nk e trớ ỗ n 1✱ e ✱ ♠ä✐ x, y ∈ Bn ♥❤➢♥❣ Bn ❦❤➠♥❣ ➤➢ỵ❝ n xn ∈ Bn ✈í✐ ♠ä✐ n ≥ 1✳ ❚❛ ❝❤Ø r❛ {xn }n≥1 d(x, y) ❦❤➠♥❣ ❝ã ❞➲② ❝♦♥ ♥➭♦ ❤é✐ tô tr♦♥❣ {xn }n≥1 c✱ ✈í✐ ♠ä✐ x, y ∈ Ba d(x, y) s❛♦ ❝❤♦ X ✳ ●✐➯ sö {xnk }k≥1 ❧➭ ♠ét ❞➲② ❝♦♥ ❝ñ❛ x ∈ X ✳ ▲✃② α0 ∈ I s❛♦ ❝❤♦ N (x, f ) ⊆ Aα0 ✳ s❛♦ ❝❤♦ x ∈ Aα0 ✳ ❱× Aα0 ❧➭ ❇➞② ❣✐ê✱ ❝❤ä♥ sè tù ♥❤✐➟♥ ✈➭ f d(xnK , x) ✈í✐ ♠ä✐ k ≥ K ❑❤✐ ➤ã d(x, t) ≤ d(x, xnk ) + d(xnk , t) ✈í✐ ♠ä✐ r❛ t ∈ BnK (t ≥ K)✳ {xn }n≥1 f BnK ⊆ N (x, f ) ⊆ Aα0 ❱× t❤Õ ❦❤➠♥❣ ❝ã ❞➲② ❝♦♥ ❤é✐ tơ tr♦♥❣ X✳ ➜✐Ị✉ ♥➭② ❝❤Ø ❚r➳✐ ✈í✐ ❣✐➯ t❤✐Õt (X, d) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ ❝♦♠♣➝❝ ❞➲②✳ ▼➞✉ t❤✉➱♥ ♥➭② ❤♦➭♥ t❤➭♥❤ ❝❤ø♥❣ ♠✐♥❤✳ ✷✳✷✳✷✳ ▼Ư♥❤ ➤Ị ❞➲②✳ ❑❤✐ ➤ã✱ X x1 , , x m ∈ X ✭❬✼❪✮✳ ❈❤♦ (X, d) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ ❝♦♠♣➝❝ ❧➭ ❤♦➭♥ t♦➭♥ ❜Þ ❝❤➷♥✱ tø❝ ❧➭ ✈í✐ ♠ä✐ s❛♦ ❝❤♦ m X= N (xi , c) i=1 ✸✵ c tå♥ t➵✐ ❈❤ø♥❣ ♠✐♥❤✳ ●✐➯ sư ♣❤➳t ❜✐Ĩ✉ tr➟♥ ❦❤➠♥❣ ➤ó♥❣✳ ❑❤✐ ➤ã✱ tå♥ t➵✐ s❛♦ ❝❤♦ c m X= N (xi , c) i=1 x1 , , xm ∈ X ✈í✐ ♠ä✐ ❉♦ ➤ã✱ ✈í✐ ❜✃t ❦ú s❛♦ ❝❤♦ x1 ∈ X tå♥ t➵✐ ❞➲② {xn }n≥1 k−1 xk ∈ X\ N (xi , c) i=1 ✈í✐ ♠ä✐ k ≥ 2✳ ◆❤➢♥❣ ➤✐Ò✉ ♥➭② ❦Ð♦ t❤❡♦ ❞➲② {xn }n≥1 ❦❤➠♥❣ ❝ã ❞➲② ❝♦♥ ❤é✐ tô tr♦♥❣ X ✳ ▼➞✉ t❤✉➱♥ ♥➭② s✉② r❛ ➤✐Ò✉ ♣❤➯✐ ❝❤ø♥❣ ♠✐♥❤✳ ✷✳✷✳✸✳ ➜Þ♥❤ ❧ý ✭❬✼❪✮✳ ●✐➯ sư (X, d) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥✳ ❑❤✐ ➤ã ❝➳❝ ♣❤➳t ❜✐Ó✉ s❛✉ ❧➭ t➢➡♥❣ ➤➢➡♥❣ ✭❛✮ X ❧➭ ❝♦♠♣➝❝✳ ✭❜✮ ▼ä✐ t❐♣ ❝♦♥ ✈➠ ❤➵♥ ❝đ❛ ✭❝✮ X X ➤Ị✉ ❝ã ➤✐Ĩ♠ ❣✐í✐ ❤➵♥ tr♦♥❣ ❧➭ ❝♦♠♣➝❝ ❞➲②✳ ❈❤ø♥❣ ♠✐♥❤✳ ✭❛✮ ⇒ ✭❜✮✳ ●✐➯ sö ep s❛♦ ❝❤♦ A ⊆ X X✳ ❦❤➠♥❣ ❝ã ➤✐Ĩ♠ ❣✐í✐ ❤➵♥ ♥➭♦ tr♦♥❣ X✳ ❧➭ ♠ét t❐♣ ❝♦♥ ✈➠ ❤➵♥ ♠➭ ❑❤✐ ➤ã✱ ✈í✐ ♠ä✐ N (p, ep ) ∩ (A\{p}) = ∅✳ ❉♦ ➤ã p ∈ X✱ tå♥ t➵✐ N (p, ep ) ∩ A ⊆ {p}✳ ❱× t❤Õ {N (p, ep )}p∈X ❧➭ ♠ét ♣❤đ ♠ë ❝đ❛ X ❦❤➠♥❣ ❝ã ♣❤đ ❝♦♥ ❤÷✉ ❤➵♥✳ ➜✐Ị✉ ♥➭② ♠➞✉ t❤✉➱♥ ✈í✐ ✭❜✮ ⇒ ✭❝✮✳ X ●✐➯ sö ❧➭ ❝♦♠♣➝❝✳ ❚õ ➤ã t❛ ❝ã ✭❜✮✳ {xn }n≥1 ❧➭ ♠ét ❞➲② ❜✃t ❦ú tr♦♥❣ ❝ã ✈➠ ❤➵♥ ♣❤➬♥ tư ♥➟♥ ♥ã ❝ã ➤✐Ĩ♠ ❣✐í✐ ❤➵♥ N (x, e) ∩ X✳ ❱× ❞➲② {xn }n≥1 x ∈ X ✳ ❑❤✐ ➤ã✱ ✈í✐ e✱ {xn }\{x} = n≥1 ❱× t❤Õ tå♥ t➵✐ d(xnk ; x) {xnk }k≥1 ❧➭ ❞➲② ❝♦♥ ❝ñ❛ {xn }n≥1 e✳ ❱❐② {xnk }k≥1 ❤é✐ tơ ✈Ị x✳ ❱❐② X ✸✶ ♠➭ xnk ∈ N (x, e) ❧➭ ❝♦♠♣➝❝ ❞➲②✳ ❚õ ➤ã ⇒ ✭❝✮✳ ●✐➯ sư {Aα }α∈I ✭❜✮ X ✳ ❚❤❡♦ ▼Ư♥❤ ➤Ị ✷✳✷✳✶✱ c s ỗ D X d(x, y) tå♥ t➵✐ α0 ∈ I ❧➭ ♠ét ♣❤ñ ♠ë ❝đ❛ c ✈í✐ ♠ä✐ x, y ∈ D✳ ❈ã D ⊆ Aα0 ✳ ❚❤❡♦ ▼Ư♥❤ ➤Ị ✷✳✷✳✷✱ tå♥ t➵✐ x1 , x2 , , xm ∈ X m c X = N xi , ✳ ❉♦ ➤ã tå♥ t➵✐ α1 , α2 , , αm ∈ X s❛♦ ❝❤♦ i=1 s❛♦ ❝❤♦ s❛♦ ❝❤♦ m X= Aαi ✳ ❱× t❤Õ X ❧➭ ❝♦♠♣➝❝✳ i=1 ✷✳✷✳✹✳ ▼Ư♥❤ ➤Ị P ➤đ✱ (X, d) ✭❬✼❪✮✳ ●✐➯ sö ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ ➤➬② ❧➭ ♥ã♥ ❝❤✉➮♥ t➽❝ ✈í✐ ❤➺♥❣ sè ❝❤✉➮♥ t➽❝ K ✈➭ t❐♣ rỗ ó ị ủ lim sup {An }n≥1 ❧➭ ❞➲② ❝➳❝ X s❛♦ ❝❤♦ d(x, y) = n→∞ x,y∈An ∞ An ❑❤✐ ➤ã ỉ ột ể n=1 ứ ỗ n ≥ 1✱ ❝❤ä♥ xn ∈ An ✳ ❚õ ❣✐➯ t❤✐Õt t❛ ❝ã lim m,n→∞ ❉♦ ➤ã ❝❤♦ d(xn , xm ) = {xn }n≥1 ❧➭ ❞➲② ❈❛✉❝❤②✳ ❱× (X, d) ❧➭ ➤➬② ➤ñ ♥➟♥ tå♥ t➵✐ x ∈ X xn → x✳ ▼➷t ❦❤➳❝✱ ❞♦ s❛♦ {An }n≥1 ❧➭ ❞➲② ❝➳❝ t❐♣ ➤ã♥❣ ❧å♥❣ ♥❤❛✉ ♥➟♥ ∞ An ✳ ●✐➯ sö x, y ∈ An ✳ ❚õ n=1 n=1 ∞ x ∈ An ✈í✐ ♠ä✐ n ≥ 1✳ ❉♦ ➤ã x ∈ lim sup n→∞ x,y∈An d(x, y) = ∞ t❛ ❝ã d(x, y) = 0✳ ❉♦ ➤ã x = y ✳ ❱❐② An ❝❤Ø ❝ã ❞✉② ♥❤✃t ♠ét ➤✐Ĩ♠✳ n=1 ✷✳✷✳✺✳ ➜Þ♥❤ ❧ý P ✭❬✼❪✮✳ ❈❤♦ (X, d) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ ➤➬② ➤ñ ✈➭ ❧➭ ♥ã♥ ❝❤✉➮♥ t➽❝ ✈í✐ ❤➺♥❣ sè ❝❤✉➮♥ t➽❝ ✈➭ ❝❤Ø ❦❤✐ K ✳ ❑❤✐ ➤ã✱ X ❧➭ ❝♦♠♣➝❝ ❦❤✐ (X, d) ❧➭ ➤➬② ➤đ ✈➭ ❤♦➭♥ t♦➭♥ ❜Þ ❝❤➷♥✳ ❈❤ø♥❣ ♠✐♥❤✳ ●✐➯ sö X ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ ❝♦♠♣➝❝ ➤➬② ➤đ✳ ❑❤✐ ✸✷ ➤ã✱ t❤❡♦ ▼Ư♥❤ ➤Ị ✷✳✷✳✸ t❛ ❝ã r❛ X X ❧➭ ❝♦♠♣➝❝ ❞➲② ✈➭ tõ ▼Ö♥❤ ➤Ị ✷✳✷✳✶ t❛ s✉② ❧➭ ❤♦➭♥ t♦➭♥ ❜Þ ❝❤➷♥✳ ➜Ĩ ứ ề ợ ủ ị ý t ❞ơ♥❣ ▼Ư♥❤ ➤Ị ✷✳✷✳✸✱ tø❝ ❧➭ ❝❤ø♥❣ ♠✐♥❤ ♠ä✐ t❐♣ ❝♦♥ ✈➠ ❤➵♥ ❝đ❛ X✳ ●✐➯ sư F ⊆X A ❧➭ t❐♣ ❝♦♥ ✈➠ ❤➵♥ ❝ñ❛ s❛♦ ❝❤♦ X ✈➭ X ➤Ị✉ ❝ã ➤✐Ĩ♠ ❣✐í✐ ❤➵♥ tr♦♥❣ c✳ ❈❤ä♥ t❐♣ ❝♦♥ ❤÷✉ ❤➵♥ (x, c)✳ ❑❤✐ ➤ã✱ tå♥ t➵✐ x1 ∈ F X= s❛♦ ❝❤♦ A ∩ B(x1 , c) x∈F ❧➭ t❐♣ ✈➠ ❤➵♥✳ ●✐➯ sö xn+1 ∈ X A ∩ B(x1 , c) ❣å♠ ❝➳❝ ♣❤➬♥ tö x1 , x2 , , xn ✳ ❈❤ä♥ s❛♦ ❝❤♦ En = A ∩ B(x1 , c) ∩ B x2 , ❧➭ t❐♣ ✈➠ ❤➵♥✳ ❱× t❤Õ t❛ ❝ã ♥❤❛✉✱ ➤ã♥❣ ✈➭ ❜Þ ❝❤➷♥ ❝đ❛ {En }n≥1 c c ∩ · · · ∩ B xn+1 , n+1 t rỗ X ✳ ❚õ ➤ã t❛ ❝ò♥❣ ❝ã lim sup n→∞ x,y∈An d(x, y) = ∞ ∞ En ❚❤❡♦ ▼Ö♥❤ ➤Ị ✷✳✷✳✹✱ n=1 ♥➭♦ ➤ã t❤✉é❝ ❣✐í✐ ❤➵♥ ❝đ❛ X t❤× d(xn , x) En = {x} ✈í✐ x ❝❤Ø ❣å♠ ♠ét ➤✐Ó♠✳ ◆Õ✉ n=1 c ❚õ ➤ã s✉② r❛ xn → x✳ ❱× t❤Õ x ❧➭ ➤✐Ĩ♠ n A✳ ✸✸ ❑Õt ❧✉❐♥ ❙❛✉ ♠ét q✉➳ tr×♥❤ ❧➭♠ ✈✐Ư❝ ✈➭ ♥❣❤✐➟♥ ❝ø✉ tÝ❝❤ ❝ù❝✱ ♥❣❤✐➟♠ tó❝✱ ❧✉❐♥ ✈➝♥ ➤➲ t ợ ết q s rì ó ệ t❤è♥❣ ❝➳❝ ❦❤➳✐ ♥✐Ư♠✿ ♥ã♥ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ➤Þ♥❤ ❝❤✉➮♥✱ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ ✈➭ ❦❤➳✐ ♥✐Ư♠ ❞➲② ❤é✐ tơ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥✳ ❈❤ø♥❣ ♠✐♥❤ ❝❤✐ t✐Õt ♠ét sè tÝ♥❤ ❝❤✃t ✈Ị ♥ã♥ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ➤Þ♥❤ ❝❤✉➮♥ ♠➭ t➭✐ ❧✐Ư✉ ❦❤➠♥❣ ❝❤ø♥❣ ♠✐♥❤✳ ✷✳ ❚r×♥❤ ❜➭②✱ ❝❤ø♥❣ ♠✐♥❤ ❝❤✐ t✐Õt ♠ét sè ➤Þ♥❤ ❧ý ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ➳♥❤ ①➵ ❝♦rót tr➟♥ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥✳ ✸✳ ❚r×♥❤ ❜➭② ❝➳❝ tÝ♥❤ ❝❤✃t t➠♣➠ ❝đ❛ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥✱ tÝ♥❤ ❝♦♠✲ ♣➝❝ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥✳ ❈❤ø♥❣ ♠✐♥❤ ❝❤✐ t✐Õt ❝➳❝ ❦Õt q✉➯ ♠➭ t➭✐ ❧✐Ö✉ ❦❤➠♥❣ ❝❤ø♥❣ ♠✐♥❤✳ ❈❤➻♥❣ ❤➵♥ ❝➳❝ ▼Ư♥❤ ➤Ị ✷✳✶✳✹✱ ✷✳✶✳✺✱ ✷✳✶✳✻✱ ✷✳✶✳✼✱ ✷✳✶✳✾ ✈➭ ➜Þ♥❤ ❧ý ✷✳✷✳✸✳ ✸✹ t➭✐ ❧✐Ư✉ t❤❛♠ ❦❤➯♦ ❬✶❪ ▼✳ ❆❜❜❛s✱ ●✳ ❏✉♥❣❝❦ ✭✷✵✵✽✮✱ ❈♦♠♠♦♥ ❢✐①❡❞ ♣♦✐♥t r❡s✉❧ts ❢♦r ♥♦♥✲ ❝♦♠♠✉t✐♥❣ ♠❛♣♣✐♥❣s ✇✐t❤♦✉t ❝♦♥t✐♥✉✐t② ✐♥ ❝♦♥❡ ♠❡tr✐❝ s♣❛❝❡s✱ ❏✳ ▼❛t❤✳ ❆♥❛❧✳ ❆♣♣❧✳✱ ✸✹✶✭✶✮✱ ✹✶✻ ✲ ✹✷✵✳ ❬✷❪ ❑✳ ❉❡✐♠❧✐♥❣ ✭✶✾✽✺✮✱ ◆♦♥❧✐♥❡❛r ❋✉♥❝t✐♦♥❛❧ ❆♥❛❧②s✐s✱ ❙♣r✐♥❣❡r✲ ❱❡r❧❛❣✳ ❬✸❪ ❍✳ ▲♦♥❣ ✲ ●✉❛♥❣✱ ❩✳ ❳✐❛♥ ✭✷✵✵✼✮✱ ❈♦♥❡ ♠❡tr✐❝ s♣❛❝❡s ❛♥❞ ❢✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠s ♦❢ ❝♦♥tr❛❝t✐✈❡ ♠❛♣♣✐♥❣s✱ ❏✳ ▼❛t❤✳ ❆♥❛❧✳ ❆♣♣❧✳✱ ✸✸✷✱ ✶✹✻✽ ✲ ✶✹✼✻✳ ❬✹❪ ❍✳ ▼♦❤❡❜✐✱ ❍✳ ❙❛❞❡❣❤✐✱ ❆✳ ▼✳ ❘✉❜✐♥♦✈ ✭✷✵✵✻✮✱ ❇❡st ❛♣♣r♦①✐♠❛t✐♦♥ ✐♥ ❛ ❝❧❛ss ♦❢ ♥♦r♠❡❞ s♣❛❝❡s ✇✐t❤ st❛r✲s❤❛♣❡❞ ❝♦♥❡✱ ◆✉♠❡r✳ ❋✉♥❝t✳ ❆♥❛❧✳ ❖♣t✐♠✳✱ ✷✼✱ ✸✲✹✱ ✹✶✶✲✹✸✻✳ ❬✺❪ ❍✳ ▼♦❤❡❜✐ ✭✷✵✵✹✮✱ ❉♦✇♥✇❛r❞ s❡ts ❛♥❞ t❤❡✐r ❜❡st s✐♠✉❧t❛♥❡♦✉s ❛♣✲ ♣r♦①✐♠❛t✐♦♥ ♣r♦♣❡rt✐❡s ✇✐t❤ ❛♣♣❧✐❝❛t✐♦♥s✱ ◆✉♠❜❡r✳ ❋✉♥❝t✳ ❆♥❛❧✳ ❖♣t✐♠✳✱ ✷✺✱ ✭✼✲✽✮✱ ✻✽✺✲✼✵✺✳ ❬✻❪ ❍✳ ▼♦❤❡❜✐✱ ❍✳ ❙❛❞❡❣❤✐ ✭✷✵✵✺✮✱ ❇❡st ❛♣♣r♦①✐♠❛t✐♦♥ ✐♥ ❛ ❝❧❛ss ♦❢ ♦r❞❡r❡❞ ♥♦r♠❡❞ s♣❛❝❡s ✇✐t❤♦✉t ❛ ♣r❡✲♦r❞❡r r❡❧❛t✐♦♥✱ ■♥t✳ ❏✳ P✉r❡ ❆♣♣✳ ▼❛t❤✳✱ ✷✱ ✶✾✾✲✷✷✶✳ ❬✼❪ ❙❤✳ ❘❡③❛♣♦✉r✱ ▼✳ ❉❡r❛❢s❤♣♦✉r✱ ❘✳ ❍❛♠❧❜❛r❛♥✐✱ ❆ ❘❡✈✐❡✇ ♦♥ ❚♦♣♦✲ ❧♦❣✐❝❛❧ Pr♦♣❡rt✐❡s ♦❢ ❈♦♥❡ ▼❡tr✐❝ ❙♣❛❝❡✱ Pr❡♣r✐♥t✳ ❬✽❪ ❇✳ ❊✳ ❘❤♦❛❞❡s ✭✶✾✼✼✮✱ ❆ ❝♦♠♣❛r✐s♦♥ ♦❢ ✈❛r✐♦✉s ❞❡❢✐♥✐t✐♦♥ ♦❢ ❝♦♥✲ tr❛❝t✐✈❡ ♠❛♣♣✐♥❣s✱ ❚r❛♥s✳ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳✱ ✷✷✻✱ ✷✺✼✲✷✾✵✳ ❬✾❪ ❲✳ ❘✉❞✐♥ ✭✶✾✾✶✮✱ ❋✉♥❝t✐♦♥❛❧ ❛♥❛❧②s✐s✱ ▼❝●r❛✇✲❍✐❧❧✱ ❙❡❝♦♥❞ ❡❞✐✲ t✐♦♥✳ ❬✶✵❪ ❉✳ ❚✉r❦♦❣❧✉ ❛♥❞ ▼✳ ❆❜✉❧♦❤❛ ✭✷✵✶✵✮✱ ❈♦♥❡ ♠❡tr✐❝ s♣❛❝❡s ❛♥❞ ❢✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠s ✐♥ ❞✐❛♠❡tr✐❝❛❧❧② ❝♦♥tr❛❝t✐✈❡ ♠❛♣♣✐♥❣s✱ ❆❝t❛ ▼❛t❤✳ ❙✐♥✐❝❛ ❙❡r✳ ❊♥❣❧✐s❤✱ ✷✻ ✭✸✮✱ ✹✽✾ ✲ ✹✾✻✳ ❬✶✶❪ ❍✳ ❑✳ ❳✉ ✭✷✵✵✹✮✱ ❉✐❛♠❡tr✐❝❛❧❧② ❝♦♥tr❛❝t✐✈❡ ♠❛♣♣✐♥❣s✱ ❇✉❧❧❡t✐♥ ♦❢ t❤❡ ❆✉str❛❧✐❛♥ ▼❛t❤❡♠❛t✐❝❛❧ s♦❝✐❡t②✱ ✼✵ ✭✸✮✱ ✹✻✸ ✲ ✹✻✽✳ ✸✺