✶ ▼Ö❈ ▲Ö❈ ❚r❛♥❣ ▼Ö❈ ▲Ö❈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ▼Ð ✣❺❯ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ❈❤÷ì♥❣ ✶✳ ❑❤ỉ♥❣ ❣✐❛♥ tü❛ ♠➯tr✐❝ ♥â♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✶✳ ▼ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✷✳ ◆â♥ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✸✳ ❑❤æ♥❣ ❣✐❛♥ tü❛ ♠➯tr✐❝ ♥â♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ữỡ ởt số ỵ sỹ tỗ t ✤✐➸♠ ❜➜t ✤ë♥❣ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ tü❛ ♠➯tr✐❝ ♥â♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỹ tỗ t↕✐ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ ❝♦ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ tü❛ ♠➯tr✐❝ ♥â♥ ✤➛② ✤õ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỹ tỗ t t ❝õ❛ →♥❤ ①↕ ❝♦ s✉② rë♥❣ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ tü❛ ♠➯tr✐❝ ♥â♥ ✤➛② ✤õ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỹ tỗ t t tr ổ ❣✐❛♥ tü❛ ♠➯tr✐❝ ♥â♥ ✈ỵ✐ ♥â♥ ▼✐♥✐❤❡❞r❛❧ ♠↕♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ❑➌❚ ▲❯❾◆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ❚⑨■ ▲■➏❯ ❚❍❆▼ ❑❍❷❖ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ỵ tt t ởt tr ỳ ữợ ự q trồ t➼❝❤ ❤➔♠✱ ♥â ❝â ♥❤✐➲✉ ù♥❣ ❞ö♥❣ tr♦♥❣ ❣✐↔✐ t➼❝❤ ✈➔ ♠ët sè ♥❣➔♥❤ t♦→♥ ❤å❝ ❦❤→❝✳ ❉♦ ✤â ♥â ✤÷đ❝ ❝→❝ ♥❤➔ t♦→♥ ❤å❝ q✉❛♥ t➙♠ ♥❣❤✐➯♥ ❝ù✉ ✈➔ t ữủ t q ỵ t ✤è✐ ✈ỵ✐ →♥❤ ①↕ ❝♦ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ✤➛② ✤õ ❝õ❛ ❇❛♥❛❝❤ ✤➣ ✤÷đ❝ ♠ð rë♥❣ ❝❤♦ ♥❤✐➲✉ ❦✐➸✉ →♥❤ ①↕ tr➯♥ ♥❤✐➲✉ ❧♦↕✐ ❦❤æ♥❣ ❣✐❛♥ ❦❤→❝ ♥❤❛✉✳ ▼ët tr ỳ ữợ rở õ ợt tr tr tứ õ t ữủ ợ ❝→❝ ❦❤ỉ♥❣ ❣✐❛♥ tü❛ ♠➯tr✐❝ rë♥❣ ❤ì♥ ❧ỵ♣ ❝→❝ ❦❤ỉ♥❣ tr õ ữớ t ự sỹ tỗ t↕✐ ❝→❝ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ ❝→❝ →♥❤ ①↕ tr➯♥ ❦❤ỉ♥❣ ❣✐❛♥ tü❛ ♠➯tr✐❝✳ ◆❤ú♥❣ ♥❣÷í✐ t❤✉ ✤÷đ❝ ♥❤✐➲✉ ❦➳t q t ữợ st ❘✳ ❆✳ ❙t♦❧t❡♥❜❡❣✱ ❈✳ ❙✳ ❲♦♥❣✳✳✳ ◆➠♠ ✷✵✵✼✱ ❍✉❛♥❣✱ ▲♦♥❣ ✲ ●✉❛♥❣ ✈➔ ❩❤❛♥❣ ❳✐❛♥ ✤➣ ✤÷❛ r❛ ❦❤→✐ ♥✐➺♠ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥ ❜➡♥❣ ❝→❝❤ t❤❛② ❣✐↔ t❤✐➳t ❤➔♠ ♠➯tr✐❝ ♥❤➟♥ ❣✐→ trà tr♦♥❣ t➟♣ ❝→❝ sè t❤ü❝ ❦❤æ♥❣ tr tr ởt õ ữợ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✳ ❈ơ♥❣ t÷ì♥❣ tü ♥❤÷ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥ ❝â t❤➸ ✤÷❛ r❛ ❦❤→✐ ♥✐➺♠ ❦❤ỉ♥❣ ❣✐❛♥ tü❛ ♠➯tr✐❝ ♥â♥ ❜➡♥❣ ❝→❝❤ t❤❛② t❤❛② ❣✐↔ t❤✐➳t ❤➔♠ tü❛ ♠➯tr✐❝ ♥❤➟♥ ❣✐→ trà tr♦♥❣ t➟♣ ❝→❝ sè t❤ü❝ ❦❤æ♥❣ ➙♠ ❜ð✐ ♥❤➟♥ ❣✐→ trà tr♦♥❣ ♠ët ♥â♥ ✤à♥❤ ữợ tr ổ t ữủt ự ú tổ t ữợ ự ♥❤➡♠ ♠ư❝ ✤➼❝❤ ❧➔ ✤÷❛ r❛ ❦❤→✐ ♥✐➺♠ ❦❤ỉ♥❣ ❣✐❛♥ tü❛ ♠➯tr✐❝ ♥â♥ ✈➔ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ t➼♥❤ ❝❤➜t tæ♣æ sỹ tỗ t t tr ổ ✤â✳ ❱ỵ✐ ♠ư❝ ✤➼❝❤ ✤â✱ ❧✉➟♥ ✈➠♥ ❝❤✐❛ ❧➔♠ ❤❛✐ ❝❤÷ì♥❣ ❈❤÷ì♥❣ ✶✳ ❑❤ỉ♥❣ ❣✐❛♥ tü❛ ♠➯tr✐❝ ♥â♥ ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔②✱ ✤➛✉ t✐➯♥ ❝❤ó♥❣ tỉ✐ ♥❤➢❝ ❧↕✐ ♠ët sè ❦❤→✐ ♥✐➺♠ ❝ì ❜↔♥ ❝õ❛ tỉ♣ỉ ✤↕✐ ❝÷ì♥❣✱ ❣✐↔✐ t➼❝❤ ❤➔♠ ❝â ❧✐➯♥ q✉❛♥ ✤➳♥ ♥ë✐ ❞✉♥❣ ❝õ❛ ❧✉➟♥ ✈➠♥✳ ❚r➻♥❤ ❜➔② ❦❤→✐ ♥✐➺♠ ♥â♥ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✱ ✈➼ ❞ư ✈➔ ❝→❝ t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ♥â♥ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✳ ✸ ❙❛✉ ✤â✱ ❝❤ó♥❣ tỉ✐ ✤÷❛ r❛ ❦❤→✐ ♥✐➺♠ ❦❤ỉ♥❣ ❣✐❛♥ tü❛ ♠➯tr✐❝ ♥â♥✱ ✈➼ ❞ư ✈➔ ❝→❝ t➼♥❤ ❝❤➜t tæ♣æ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ tü❛ ♠➯tr✐❝ ♥â♥ ♠➔ ❝❤ó♥❣ ❝➛♥ ❞ị♥❣ tr♦♥❣ ❝❤÷ì♥❣ ✷✳ ❈❤÷ì♥❣ ✷✳ ▼ët sè ỵ sỹ tỗ t t tr ❦❤ỉ♥❣ ❣✐❛♥ tü❛ ♠➯tr✐❝ ♥â♥ ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔②✱ ❝❤ó♥❣ tỉ✐ ự sỹ tỗ t t →♥❤ ①↕ ❝♦ ✈➔ →♥❤ ①↕ ❝♦ s✉② rë♥❣ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ tü❛ ♠➯tr✐❝ ♥â♥✳ ✣➛✉ t✐➯♥✱ ❝❤ó♥❣ tỉ✐ ❝❤ù♥❣ t q tữỡ tỹ ữ ỵ sỹ tỗ t t →♥❤ ①↕ ❝♦ s✉② rë♥❣ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ tü❛ ♠➯tr✐❝ ✈➔ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥ ✈➝♥ ✤ó♥❣ ❝❤♦ ❦❤ỉ♥❣ ❣✐❛♥ tü❛ ♠➯tr✐❝ ♥â♥✳ ❙❛✉ ✤â ❝❤ó♥❣ tỉ✐ ✤÷❛ r❛ ♠ët số ỵ sỹ tỗ t t ❝õ❛ ❝→❝ →♥❤ ①↕ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ tü❛ ♠➯tr✐❝ ♥â♥ ợ õ r ữủ t t rữớ ữợ sỹ ữợ t t ❝❤✉ ✤→♦ ✈➔ ♥❣❤✐➯♠ ❦❤➢❝ ❝õ❛ t❤➛② ❣✐→♦ P●❙✳❚❙✳ ✣✐♥❤ ❍✉② ❍♦➔♥❣✳ ❚→❝ ❣✐↔ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ ♥❤➜t ❝õ❛ ♠➻♥❤ ✤➳♥ ❚❤➛②✱ ♥❣÷í✐ ✤➣ ❝❤➾ ❞↕② t→❝ ❣✐↔ ♥❤ú♥❣ ❦✐➳♥ t❤ù❝✱ ❦✐♥❤ ♥❣❤✐➺♠ tr♦♥❣ ❤å❝ t➟♣ ✈➔ ♥❣❤✐➯♥ ❝ù✉ ❦❤♦❛ ❤å❝✳ ❚→❝ ❣✐↔ ①✐♥ ✤÷đ❝ ỡ qỵ ổ t tr♦♥❣ ❑❤♦❛ ❚♦→♥ ✲ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❱✐♥❤ ✤➣ ♥❤✐➺t t➻♥❤ ❣✐↔♥❣ ❞↕② ✈➔ ❣✐ó♣ ✤ï t→❝ ❣✐↔ tr♦♥❣ s✉èt t❤í✐ ❣✐❛♥ ❤å❝ t➟♣✳ ❈✉è✐ ❝ị♥❣ ①✐♥ ❝↔♠ ì♥ ❣✐❛ ỗ t tr♦♥❣ ❧ỵ♣ ❈❛♦ ❤å❝ ✶✽ ✲ ❈❤✉②➯♥ ♥❣➔♥❤✿ ●✐↔✐ t➼❝❤ ✤➣ ❝ë♥❣ t→❝✱ ❣✐ó♣ ✤ï ✈➔ ✤ë♥❣ ✈✐➯♥ t→❝ ❣✐↔ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ ♥❣❤✐➯♥ ❝ù✉✳ ▼➦❝ ❞ị ✤➣ ❝â ♥❤✐➲✉ ❝è ❣➢♥❣ ♥❤÷ ❞♦ ❝á♥ ❤↕♥ ❝❤➳ ✈➲ ♠➦t ❦✐➳♥ t❤ù❝ ✈➔ t❤í✐ ❣✐❛♥ ♥➯♥ ❧✉➟♥ ổ tr ọ ỳ t sõt qỵ ổ õ õ ỵ ❧✉➟♥ ✈➠♥ ✤÷đ❝ ❤♦➔♥ t❤✐➺♥ ❤ì♥✳ ❱✐♥❤✱ t❤→♥❣ ✵✾ ♥➠♠ ✷✵✶✷ ❚→❝ ❣✐↔ ✹ ❈❍×❒◆● ✶ ❑❍➷◆● ●■❆◆ ❚Ü❆ ▼➊❚❘■❈ ◆➶◆ ✶✳✶✳ ▼❐❚ ❙➮ ❑■➌◆ ❚❍Ù❈ ❈❍❯❽◆ ❇➚ ▼ö❝ ♥➔② ❞➔♥❤ ❝❤♦ ✈✐➺❝ ❣✐ỵ✐ t❤✐➺✉ ❝→❝ ❦❤→✐ ♥✐➺♠ ❝ì ❜↔♥ ✈➔ ❝→❝ ❦➳t q✉↔ ✤➣ ❝â ❝➛♥ ❞ò♥❣ tr♦♥❣ ❧✉➟♥ ✈➠♥✳ ✶✳✶✳✶✳ ✣à♥❤ ♥❣❤➽❛ ✭❬✶❪✮✳ ❈❤♦ t➟♣ ❤ñ♣ X ✳ ❍å τ ❝→❝ t➟♣ ❝♦♥ ❝õ❛ X ✤÷đ❝ ❣å✐ ❧➔ tæ♣æ tr➯♥ X ♥➳✉ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ (T1 ) φ, X ∈ τ ❀ (T2 ) ◆➳✉ Gi ∈ τ ∈ I t❤➻ Gi ∈ τ ❀ i∈I ◆➳✉ G1, G2 ∈ τ t❤➻ G1 ∩ G2 ∈ τ ✳ ❚➟♣ ❤đ♣ X ❝ị♥❣ ✈ỵ✐ tỉ♣ỉ τ tr➯♥ ♥â ữủ ổ tổổ ỵ (X, τ ) ❤❛② ✤ì♥ ❣✐↔♥ X ✳ ❈→❝ ♣❤➛♥ tû ❝õ❛ X ✤÷đ❝ ❣å✐ ❧➔ ✤✐➸♠ ❝õ❛ ❦❤ỉ♥❣ ❣✐❛♥ tỉ♣ỉ✳ ❈→❝ ♣❤➛♥ tû t❤✉ë❝ τ ✤÷đ❝ ❣å✐ ❧➔ t➟♣ ♠ð✳ ●✐↔ sû E ⊂ X ✳ ❚➟♣ E ✤÷đ❝ ❣å✐ ❧➔ t➟♣ ✤â♥❣ ♥➳✉ X\E ❧➔ t➟♣ ♠ð✳ ✶✳✶✳✷✳ ✣à♥❤ ♥❣❤➽❛ ✭❬✶❪✮✳ ❈❤♦ ❦❤æ♥❣ ❣✐❛♥ tæ♣æ X ✱ t➟♣ ❝♦♥ A ❝õ❛ X ✤÷đ❝ ❣å✐ ❧➔ ❧➙♥ ❝➟♥ ❝õ❛ x X tỗ t t V ⊂ X s❛♦ ❝❤♦ x ∈ V ⊆ A✳ ❈❤♦ ❦❤æ♥❣ ❣✐❛♥ tæ♣æ X ✱ x ∈ X, U(x) ❧➔ ❤å t➜t ❝↔ ❝→❝ ❧➙♥ ❝➟♥ ❝õ❛ x✳ ❍å B(x) ⊂ U(x) ✤÷đ❝ ❣å✐ ❧➔ ❝ì sð ❧➙♥ ❝➟♥ t↕✐ x ợ U U(x) tỗ t V ∈ B(x) s❛♦ ❝❤♦ V ⊂ U ✳ ✶✳✶✳✸✳ ✣à♥❤ ♥❣❤➽❛ ✭❬✶❪✮✳ ❉➣② {xn} tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ tỉ♣ỉ X ✤÷đ❝ ❣å✐ ❧➔ ❤ë✐ tư tỵ✐ x ∈ X ♥➳✉ ✈ỵ✐ ộ U x tỗ t n0 N s❛♦ ❝❤♦ xn ∈ U ✈ỵ✐ ♠å✐ n ≥ n0 ❑❤✐ ✤â✱ t❛ ✈✐➳t xn−→x ❤♦➦❝ n→∞ lim xn = x✳ (T3 ) ✶✳✶✳✹✳ ✣à♥❤ ♥❣❤➽❛ ✭❬✷❪✮✳ ❑❤æ♥❣ ❣✐❛♥ tỉ♣ỉ X ✤÷đ❝ ❣å✐ ❧➔ t❤ä❛ ♠➣♥ t✐➯♥ ✤➲ ✤➳♠ ✤÷đ❝ t❤ù ♥❤➜t ♥➳✉ t↕✐ ♠é✐ ✤✐➸♠ x ∈ X ❝â ♠ët ❝ì sð ❧➙♥ ❝➟♥ ✺ ❝â ❧ü❝ ❧÷đ♥❣ ✤➳♠ ✤÷đ❝✳ ❑❤ỉ♥❣ ❣✐❛♥ tỉ♣ỉ X ✤÷đ❝ ❣å✐ ❧➔ T2✲❦❤æ♥❣ ❣✐❛♥ ❤❛② ❦❤æ♥❣ ❣✐❛♥ ❍❛✉s✲ ❞♦r❢❢ ♥➳✉ ❤❛✐ ✤✐➸♠ t ý x, y X, x = y tỗ t↕✐ ❝→❝ ❧➙♥ ❝➟♥ t÷ì♥❣ ù♥❣ Ux , Uy ❝õ❛ x ✈➔ y s❛♦ ❝❤♦ Ux ∩ Uy = ∅✳ ◆➳✉ X ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❍❛✉s❞♦r❢❢ t❤➻ ♠é✐ ❞➣② tr♦♥❣ X ♠➔ ❤ë✐ tư t❤➻ ❤ë✐ tư tỵ✐ ♠ët ✤✐➸♠ ❞✉② ♥❤➜t✳ ✶✳✶✳✺✳ ✣à♥❤ ♥❣❤➽❛ ✭❬✷❪✮✳ ●✐↔ sû X, Y ❧➔ ❤❛✐ ❦❤æ♥❣ ❣✐❛♥ tæ♣æ ✈➔ f : X−→Y ✳ ⑩♥❤ ①↕ f ✤÷đ❝ ❣å✐ ❧➔ ❧✐➯♥ tư❝ t↕✐ x ∈ X ♥➳✉ ✈ỵ✐ ♠é✐ ❧➙♥ ❝➟♥ V ❝õ❛ f (x) tỗ t U x s f (U ) ⊂ V ✳ ⑩♥❤ ①↕ f ✤÷đ❝ ❣å✐ ❧➔ ❧✐➯♥ tö❝ tr➯♥ X ✭♥â✐ ❣å♥ ❧➔ ❧✐➯♥ tö❝✮ ♥➳✉ ♥â ❧✐➯♥ tö❝ t↕✐ ♠å✐ ✤✐➸♠ ❝õ❛ X ỵ sỷ X Y ❧➔ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ tæ♣æ✱ f : B(x) X−→Y ✳ ❑❤✐ ✤â ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ t÷ì♥❣ ✤÷ì♥❣ ✭✶✮ f ❧✐➯♥ tö❝ tr➯♥ X ❀ ✭✷✮ ◆➳✉ E ❧➔ t➟♣ ♠ð tr♦♥❣ Y t❤➻ f −1 (E) ♠ð tr♦♥❣ X ❀ ✭✸✮ ◆➳✉ E ❧➔ t➟♣ ✤â♥❣ tr♦♥❣ Y t❤➻ f −1 (E) ✤â♥❣ tr♦♥❣ X ✳ ✶✳✶✳✼✳ ✣à♥❤ ♥❣❤➽❛ ✭❬✶❪✮✳ ●✐↔ sû X ❧➔ t➟♣ ❦❤→❝ ré♥❣ ✈➔ d : X ì XR d ữủ ởt ♠➯tr✐❝ tr➯♥ X ♥➳✉ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ ✤÷đ❝ t❤ä❛ ♠➣♥ ✐✮ d(x, y) ≥ ✈ỵ✐ ♠å✐ x, y ∈ X ✈➔ d(x, y) = ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ x = y❀ ✐✐✮ d(x, y) = d(y, x) ✈ỵ✐ ♠å✐ x, y ∈ X ❀ ✐✐✐✮ d(x, y) ≤ d(x, z) + d(z, y) ✈ỵ✐ ♠å✐ x, y, z ∈ X ✳ ❚➟♣ X ❝ị♥❣ ✈ỵ✐ ♠ët ♠➯tr✐❝ tr õ ữủ ổ tr ỵ ❤✐➺✉ ❧➔ (X, d) ❤♦➦❝ X ✳ ✶✳✶✳✽✳ ✣à♥❤ ♥❣❤➽❛ ✭❬✸❪✮✳ ●✐↔ sû X ❧➔ t➟♣ ❦❤→❝ ré♥❣ ✈➔ d : X ì XR d ữủ ởt tü❛ ♠➯tr✐❝ tr➯♥ X ♥➳✉ t❤ä❛ ♠➣♥ ✐✮ d(x, y) ≥ ✈ỵ✐ ♠å✐ x, y ∈ X ✈➔ d(x, y) = ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ x = y❀ ✐✐✮ d(x, z) ≤ d(x, y) + d(y, z) ✈ỵ✐ ♠å✐ x, y, z ∈ X ✳ ❚➟♣ X ❝ò♥❣ ợ ởt tỹ tr tr õ ữủ ởt ổ tỹ tr ỵ (X, d) ❤♦➦❝ X ✳ ✶✳✶✳✾✳ ✣à♥❤ ♥❣❤➽❛ ✭❬✶❪✮✳ ●✐↔ sû E ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì tr➯♥ tr÷í♥❣ K = R ❤♦➦❝ K = C✳ ❍➔♠ p : E−→R t❤ã❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ ✐✮ p(x) ≥ 0, ∀x ∈ E ✈➔ p(x) = ⇔ x = 0❀ ✐✐✮ p(xλ) = |λ|p(x)✱ ∀x ∈ E ✱ ∀λ ∈ K❀ ✐✐✐✮ p(x + y) ≤ p(x) + p(y) ∀x, y ∈ E ✤÷đ❝ ❣♦à ❧➔ ♠ët ❝❤✉➞♥ tr➯♥ ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì E ✳ ❙è p(x) ✤÷đ❝ ❣å✐ ❧➔ ❝❤✉➞♥ ❝õ❛ ✈❡❝tì X ∈ E ✳ ❚❛ t❤÷í♥❣ ❦➼ ❤✐➺✉ ❝❤✉➞♥ ❝õ❛ x ❧➔ x ✳ ❑❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì E ❝ị♥❣ ✈ỵ✐ ♠ët ❝❤✉➞♥ ①→❝ ✤à♥❤ tr➯♥ ♥â ✤÷đ❝ ❣å✐ ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥✳ ✶✳✶✳✶✵✳ ▼➺♥❤ ✤➲ ✭❬✶❪✮✳ ◆➳✉ E ❧➔ ❦❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥ t❤➻ ❝æ♥❣ t❤ù❝ d(x, y) = x − y , ∀x, y ∈ E ①→❝ ✤à♥❤ ♠ët ♠➯tr✐❝ tr➯♥ E ✳ ❚❛ ❣å✐ ♠➯tr✐❝ ♥➔② ❧➔ ♠➯tr✐❝ s✐♥❤ ❜ð✐ ❝❤✉➞♥ ❤❛② ♠➯tr✐❝ ❝❤✉➞♥✳ ✶✳✶✳✶✶✳ ✣à♥❤ ♥❣❤➽❛ ✭❬✶❪✮✳ ▼ët ❦❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥ ✈➔ ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ✤➛② ✤õ t❤❡♦ ♠➯tr✐❝ s✐♥❤ ❜ð✐ ❝❤✉➞♥ t❤➻ ữủ ởt ổ ỵ ✭❬✶❪✮✳ ◆➳✉ E ❧➔ ❦❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥ t❤➻ ✶✮ →♥❤ ①↕ ❝❤✉➞♥✿ x → x , ∀x ∈ E ❀ ✷✮ ♣❤➨♣ ❝ë♥❣ ✿(x, y) → x + y, (x, y) E ì E ợ ổ ữợ (, x) x, (, x) K × E ❧➔ ❝→❝ →♥❤ ①↕ ❧✐➯♥ tư❝✳ ✶✳✶✳✶✸✳ ỵ sỷ E ổ ❝❤✉➞♥✳ ❑❤✐ ✤â✱ ✈ỵ✐ ♠é✐ a ∈ E ✈➔ ♠é✐ λ ∈ K, λ = ❝→❝ →♥❤ ①↕ x → x + a, x → λx ∀x ∈ E ỗ ổ E E ◆➶◆ ❚❘❖◆● ❑❍➷◆● ●■❆◆ ❇❆◆❆❈❍ ✼ ✶✳✷✳✶✳ ✣à♥❤ ♥❣❤➽❛ ✭❬✹❪✮✳ ❈❤♦ E ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ tr➯♥ tr÷í♥❣ sè t❤ü❝ R✳ ❚➟♣ ❝♦♥ P ❝õ❛ E ✤÷đ❝ ❣å✐ ❧➔ ♠ët ♥â♥ ♥➳✉ t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ ✐✮ P ❧➔ t➟♣ ✤â♥❣ ❦❤→❝ ré♥❣ ✈➔ P = {0}❀ ✐✐✮ ❱ỵ✐ ♠å✐ x, y ∈ P ✱ ♠å✐ a, b ∈ R, a, b ≥ t❛ ❝â ax + by ∈ P ❀ ✐✐✐✮ ◆➳✉ x ∈ P ✈➔ −x ∈ P t❤➻ x = 0✳ ✶✳✷✳✷✳ ❱➼ ❞ö ✭❬✹❪✮✳ ✶✮ ❚r♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❝→❝ sè t❤ü❝ R ✈ỵ✐ ❝❤✉➞♥ t❤ỉ♥❣ t❤÷í♥❣✱ t➟♣ P = {x ∈ R : x ≥ 0} ❧➔ ♠ët ♥â♥✳ ✷✮ ●✐↔ sû E = R2, P = {(x, y) ∈ E : x, y ≥ 0} ⊂ R2✳ ❑❤✐ ✤â✱ P t❤ä❛ ♠➣♥ ❜❛ ✤✐➲✉ ❦✐➺♥✿ ✐✮ P ❧➔ t➟♣ ✤â♥❣✱ P = ∅, P = {0}❀ ✐✐✮ ❱ỵ✐ ♠å✐ (x, y), (u, v) ∈ P ✈➔ ♠å✐ a, b ∈ R, a, b ≥ t❛ ❝â a(x, y) + b(u, v) ∈ P ❀ ✐✐✐✮ ❱ỵ✐ (x, y) ∈ P ✈➔ (−x, −y) ∈ P t❛ ❝â (x, y) = (0, 0)✳ ❱➟② P ❧➔ ♠ët ♥â♥ tr➯♥ E ✳ ❈❤♦ P ❧➔ ♠ët ♥â♥ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ E ✳ ❑❤✐ ✤â✱ tr➯♥ E ①➨t q✉❛♥ ❤➺ t❤ù tü ✧≤✧ ①→❝ ✤à♥❤ ❜ð✐ P ♥❤÷ s❛✉✿ x ≤ y ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ y − x ∈ P ✳ ❈❤ó♥❣ t q ữợ x < y x y ✈➔ x = y ✱ ❝á♥ x y ♥➳✉ y − x ∈ intP ✈ỵ✐ intP ❧➔ ♣❤➛♥ tr♦♥❣ ❝õ❛ P ✳ ✶✳✷✳✸✳ ✣à♥❤ ♥❣❤➽❛ ✭❬✹❪✮✳ ❈❤♦ P ❧➔ ♠ët ♥â♥ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ E✳ ✶✮ ◆â♥ P ✤÷đ❝ õ t tỗ t số tỹ K > s❛♦ ❝❤♦ ✈ỵ✐ ♠å✐ x, y ∈ E ✈➔ ≤ x ≤ y t❛ ❝â x ≤ K y ✳ ❙è t❤ü❝ ❞÷ì♥❣ K ♥❤ä ♥❤➜t t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ♥➔② ✤÷đ❝ ❣å✐ ❧➔ ❤➡♥❣ sè ❝❤✉➞♥ t➢❝ ❝õ❛ P ✳ ✷✮ ◆â♥ P ✤÷đ❝ ❣å✐ ❧➔ ♥â♥ ❝❤➼♥❤ q✉② ♥➳✉ ♠å✐ ❞➣② t➠♥❣ ✈➔ ❜à ❝❤➦♥ tr➯♥ tr♦♥❣ E ✤➲✉ ❤ë✐ tö✱ ♥❣❤➽❛ ❧➔✱ ♥➳✉ {xn} ❧➔ ❞➣② tr♦♥❣ E s❛♦ ❝❤♦ x1 ≤ x2 ≤ · · · ≤ xn ≤ · · · y ợ y E t tỗ t x ∈ E s❛♦ ❝❤♦ xn − x −→0 ❦❤✐ n ỵ s õ ố q ❣✐ú❛ ♥â♥ ❝❤➼♥❤ q✉② ✈➔ ♥â♥ ❝❤✉➞♥ t➢❝✳ ✶✳✷✳✹✳ ✣à♥❤ ỵ õ q tr ổ ❧➔ ✽ ♥â♥ ❝❤✉➞♥ t➢❝✳ ●✐↔ sû P ❧➔ ♠ët ♥â♥ ❝❤➼♥❤ q✉② ♥❤÷♥❣ ❦❤ỉ♥❣ ♣❤↔✐ ❧➔ ♥â♥ ❝❤✉➞♥ t➢❝✳ õ ợ ộ n t ữủ tn, sn ∈ P s❛♦ ❝❤♦ tn −sn ∈ E ✈➔ n2 tn < sn ❱ỵ✐ ♠é✐ n ≥ 1✱ ✤➦t yn = tt ✈➔ xn = st ✳ ❚❛ ❝â yn, xn, yn − xn ∈ E, yn = ✈➔ n2 ≤ xn ✈ỵ✐ ♠å✐ n ≥ 1✳ ❱➻ ∞ ∞ ∞ y y ❝❤✉é✐ = n n ❤ë✐ tö ♥➯♥ ❝❤✉é✐ n ❤ë✐ tö tr♦♥❣ E ✳ ❚ø P ❈❤ù♥❣ ♠✐♥❤✳ n n n n n 2 n=1 n n=1 ✤â♥❣ s✉② r❛ tỗ t y P s ộ ①→❝ ✤à♥❤ ❝õ❛ ❝❤✉é✐ tr➯♥ s✉② r❛ ≤ x1 ≤ x1 + n=1 ∞ yn n2 n=1 = y ✳ ❇➙② ❣✐í✱ tø xn ≤ yn x2 x2 x3 ≤ x + + ≤ · · · ≤ y 22 22 32 ❱➻ P ❧➔ ♥â♥ ❝❤➼♥❤ q✉② ♥➯♥ ❝❤✉é✐ lim n→∞ ∞ n=1 xn n2 ❤ë✐ tö✳ ❙✉② r❛ xn = 0, n2 ✤✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥ ✈ỵ✐ xnn2 ≥ ✈ỵ✐ ♠å✐ n = 1, 2, ❱➟② P ❧➔ ♥â♥ ❝❤✉➞♥ t➢❝✳ ✶✳✷✳✺✳ t ữủ ỵ ❧➔ ❦❤ỉ♥❣ ✤ó♥❣✱ tù❝ ❧➔ ❝â ♥❤ú♥❣ ♥â♥ ❝❤✉➞♥ t➢❝ ♥❤÷♥❣ ❦❤ỉ♥❣ ❝❤➼♥❤ q✉②✳ ❚❤➟t ✈➟②✱ ①➨t ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ E = CR[0, 1] ✈ỵ✐ ❝❤✉➞♥ sup : f = sup |f (x)|✳ x∈[0,1] ✣➦t P = {f ∈ E : f ≥ 0}✳ ❑❤✐ ✤â✱ P ❧➔ ♠ët ♥â♥ ✈ỵ✐ ❤➡♥❣ sè ❝❤✉➞♥ t➢❝ K = 1✳ ❚❤➟t ✈➟②✱ ❣✐↔ sû f, g ∈ E ✈➔ ≤ f ≤ g ✳ ❑❤✐ ✤â✱ ≤ f (x) ≤ g(x) ✈ỵ✐ ♠å✐ x ∈ [0, 1] ✈➔ t❛ ❝â f = sup |f (x)| = sup f (x) ≤ sup g(x) = sup |g(x)| = g , x∈[0,1] x∈[0,1] x∈[0,1] x∈[0,1] ❝❤ù♥❣ tä P ❧➔ ♥â♥ ❝❤✉➞♥ t➢❝✳ ❇➙② ❣✐í✱ t❛ ❝❤ù♥❣ ♠✐♥❤ P ❦❤ỉ♥❣ ♣❤↔✐ ❧➔ ♥â♥ ❝❤➼♥❤ q✉②✳ ❚❤➟t ✈➟②✱ ❧➜② ❞➣② {fn} tr♦♥❣ E ❝❤♦ ❜ð✐ fn(x) = xn ✈ỵ✐ ♠å✐ x ∈ [0, 1]✳ ❑❤✐ ✤â✱ t❛ ❝â ≤ · · · ≤ xn ≤ xn−1 ≤ · · · ≤ x2 ≤ x ✈ỵ✐ ♠å✐ x ∈ [0, 1] ✾ ❘ã r➔♥❣ {fn} ữợ ữ {fn} ổ ❤ë✐ tư tr♦♥❣ E ✳ ❱➟② P ❦❤ỉ♥❣ ♣❤↔✐ ❧➔ ♥â♥ ❝❤➼♥❤ q✉②✳ ✶✳✷✳✻✳ ▼➺♥❤ ✤➲ ✭❬✹❪✮✳ ◆➳✉ K ❧➔ ❤➡♥❣ sè ❝❤✉➞♥ t➢❝ ❝õ❛ ♥â♥ P t❤➻ K ≥ 1✳ ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû P ❧➔ ♠ët ♥â♥ ✈ỵ✐ ❤➡♥❣ sè ❝❤✉➞♥ t➢❝ K < 1✳ ❈❤å♥ x ∈ P, x = ✈➔ < ε < − K s❛♦ ❝❤♦ K < − ε✳ ❑❤✐ ✤â✱ (1 − ε)x ≤ x ♥❤÷♥❣ (1 − ε) x > K x ✣✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥ ✈ỵ✐ K ❧➔ ❤➡♥❣ sè ❝❤✉➞♥ t➢❝ ❝õ❛ P ✳ ❱➟② K ≥ 1✳ ✶✳✸✳ ❑❍➷◆● ●■❆◆ ❚Ü❆ ▼➊❚❘■❈ ◆➶◆ ❚r♦♥❣ t q ữợ P ởt õ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ t❤ü❝ E s❛♦ ❝❤♦ intP = φ ✈➔ q✉❛♥ ❤➺ t❤ù tü ✧≤✧ tr➯♥ E ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ P ✳ ✶✳✸✳✶✳ ✣à♥❤ ♥❣❤➽❛✳ ●✐↔ sû X ❧➔ t➟♣ ❦❤→❝ ré♥❣ ✈➔ d : X × X−→E ✳ ❍➔♠ d ✤÷đ❝ ❣å✐ ❧➔ tü❛ ♠➯tr✐❝ ♥â♥ tr➯♥ X ♥➳✉ t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ ✐✮ d(x, y) ≥ ✈ỵ✐ ♠å✐ x, y ∈ X ❀ ✐✐✮ d(x, y) = ⇔ x = y❀ ✐✐✐✮ d(x, y) ≤ d(x, z) + d(z, y) ✈ỵ✐ ♠å✐ x, y, z ∈ X ✳ ❚➟♣ X ❝ò♥❣ ợ tỹ tr õ d tr X ữủ ổ tỹ tr õ ữủ ỵ (X, d) ❤♦➦❝ X ✳ ❑❤æ♥❣ ❣✐❛♥ tü❛ ♠➯tr✐❝ ♥â♥ X ♥➳✉ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ d(x, y) = d(y, x) ợ x, y X ữủ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥ X ✳ ✶✳✸✳✷✳ ❱➼ ❞ư✳ ✶✮ ▼å✐ tü❛ ♠➯tr✐❝ ❧➔ tü❛ ♠➯tr✐❝ ♥â♥✱ ♠å✐ ❦❤æ♥❣ ❣✐❛♥ tü❛ ♠➯tr✐❝ ❧➔ ❦❤æ♥❣ ❣✐❛♥ tü❛ ♠➯tr✐❝ ♥â♥✳ ✷✮ ▼å✐ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥ ❧➔ tü❛ ♠➯tr✐❝ ♥â♥✳ ✸✮ ●✐↔ sû P = {f ∈ C[0,1] : f (x) ≥ ∀x ∈ [0, 1]}✳ ❑❤✐ ✤â✱ P ❧➔ ♥â♥ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ C[0,1] ✈ỵ✐ ❝❤✉➞♥ s✉♣✳ ▲➜② f0 ∈ P, f0 = ✈➔ ①→❝ ✶✵ ✤à♥❤ ❤➔♠ d : R × R → C[0,1] ❜ð✐ ❝ỉ♥❣ t❤ù❝ d(x, y) = f 2f0 0 ♥➳✉ ♥➳✉ ♥➳✉ x n1, j > n2, , j > nm ✈➔ ϕ(xj ) c✳ ❉♦ ✤â xj ∈ An ✳ ❚ø ✤â j=1 s✉② r❛ ❞➣② {An} ❝â t➼♥❤ ❣✐❛♦ ❤ú✉ ❤↕♥✳ ❱ỵ✐ ♠é✐ b intP (xn) tỗ t↕✐ n0 ∈ N s❛♦ ❝❤♦ ✈ỵ✐ ♠å✐ n ≥ n0 ❝â 0 j j j 1 ϕ(xn ) < µ(b) = inf{ϕ(x) + ϕ(y) : d(x, y) ≥ b} 2 ❚❤❡♦ ❝→❝❤ ①→❝ ✤à♥❤ An t❤➻ ✈ỵ✐ ♠å✐ n ≥ n0 ✈➔ x, y ∈ An ❝â ϕ(x) + ϕ(y) ≤ 2ϕ(xn ) < µ(b) ❉♦ ✤â✱ tø ❝→❝❤ ①→❝ ✤à♥❤ µ(b) s✉② r❛ d(x, y) < b✳ ✣✐➲✉ ♥➔② ❝❤ù♥❣ tä δ(An ) < b ✈ỵ✐ ♠å✐ n ≥ n0 ✳ ❚ø b ❧➔ ♣❤➛♥ tû ❜➜t ❦ý tr♦♥❣ intP s✉② r❛ ∞ An ✳ (An ) ỵ t tỗ t t u n=1 c1 intP ✳ ❱➻ u ∈ An ✈ỵ✐ ♠å✐ n ≥ ♥➯♥ c1 B u, ∩ An = ∅ ✈ỵ✐ ♠å✐ n n õ tỗ t {yn} s ❝❤♦ yn ∈ B u, c1 ∩ An , ∀n ≥ n ❚❛ ❝â d(u, xn ) ≤ d(u, yn ) + d(yn , xn ) c1 + δ(An ), ∀n ≥ n ❚ø δ(An) → ❦❤✐ n → ∞ ✈➔ ❍➺ q✉↔ ✶✳✸✳✼ s✉② r❛ d(u, xn) → 0✱ tù❝ ❧➔ xn → u✳ ✷✽ ●✐↔ sỷ d tử tỗ t v X s❛♦ ❝❤♦ ♥➳✉ {yn} ❧➔ ❞➣② tr♦♥❣ X ♠➔ ϕ(yn) → t❤➻ yn → v✳ ❑❤✐ ✤â✱ ✈➻ d ❧✐➯♥ tö❝ ♥➯♥ d(xn, yn) → d(u, v)✳ ▼➦t ❦❤→❝ ϕ(xn) + ϕ(yn) → ♥➯♥ ✈ỵ✐ ♠å✐ c ∈ intP tỗ t n0 N s ợ n ≥ n0 t❛ ❝â ϕ(xn ) + ϕ(yn ) < µ(c) = inf{ϕ(x) + ϕ(y) : c ≤ d(x, y)} ❚ø ✤â s✉② r❛ d(xn, yn) < c ✈ỵ✐ ♠å✐ n ≥ no✳ ❱➻ c ❧➔ ♣❤➛♥ tû ❜➜t ❦ý tr♦♥❣ intP ♥➯♥ tø ❍➺ q✉↔ ✶✳✸✳✼ s✉② r❛ d(xn , yn ) → 0✳ ❉♦ ✤â d(u, v) = u = v ỵ sû (X, d) ❧➔ ❦❤æ♥❣ ❣✐❛♥ tü❛ ♠➯tr✐❝ ♥â♥ ✤➛② ✤õ ❍❛✉s❞♦r❢❢ ✈➔ f : X−→X ❧➔ ❤➔♠ ❧✐➯♥ tö❝ s❛♦ ❝❤♦ ✤✐➸♠ tr♦♥❣ E ❧➔ ✤✐➸♠ ❣✐ỵ✐ ❤↕♥ ❝õ❛ t➟♣ {d(f x, x) : x ∈ X}✳ ❑❤✐ ✤â✱ ♥➳✉ ϕ(x) = d(f x, x), ∀x ∈ X tọ tr ỵ ✷✳✸✳✺ t❤➻ f ❝â ✤✐➸♠ ❜➜t ✤ë♥❣✳ ❍ì♥ ♥ú❛✱ ♥➳✉ t❤➯♠ ❣✐↔ t❤✐➳t d ❧✐➯♥ tö❝ t❤➻ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ f ❧➔ ❞✉② ♥❤➜t✳ ❈❤ù♥❣ ♠✐♥❤✳ ❚ø ❣✐↔ t❤✐➳t ✤✐➸♠ ∈ E ❧➔ ✤✐➸♠ ❣✐ỵ✐ ❤↕♥ ❝õ❛ t➟♣ {d(f x, x) : x X} s r tỗ t↕✐ ❞➣② {xn } ⊂ X s❛♦ ❝❤♦ d(f xn , xn ) → ❦❤✐ n → ∞✳ ❉♦ ✤â ϕ(xn ) = d(f xn , xn ) → ỵ tỗ t u X s❛♦ ❝❤♦ xn → u✳ ❱➻ f ❧✐➯♥ tö❝ ♥➯♥ f xn f u ỵ t d(f u, f xn ) → 0✳ ▼➦t ❦❤→❝✱ t❤❡♦ ❜➜t ✤➥♥❣ t❤ù❝ t❛♠ ❣✐→❝ t❛ ❝â d(f u, xn ) ≤ d(f u, f xn ) + d(f xn , xn ) ✈ỵ✐ ♠å✐ n ❚ø d(f u, f xn) → 0, d(f xn, xn) → 0✱ →♣ ❞ö♥❣ ❍➺ q✉↔ ✶✳✸✳✼ t❛ ❝â d(f u, xn) → 0✳ ỵ t xn f u X ❧➔ T2 ✲❦❤æ♥❣ ❣✐❛♥ ♥➯♥ u = f u✳ ❱➟② u ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ f ✳ ❇➙② ❣✐í✱ ❣✐↔ sû t❤➯♠ d ❧✐➯♥ tư❝ ✈➔ v ❝ơ♥❣ ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ f ✳ ❑❤✐ ✤â✱ tø d ✈➔ f ❧✐➯♥ tö❝ s✉② r❛ ϕ ❧✐➯♥ tö❝✳ ▲➜② ❞➣② {vn} ⊂ X, → v✳ ❱➻ ✷✾ ϕ ❧✐➯♥ tö❝ ♥➯♥ ϕ(vn ) → ϕ(v) = d(f v, v) = d(v, v) = 0✳ ❉♦ ✤â t❤❡♦ ỵ u X T2ổ ❣✐❛♥ ♥➯♥ u = v ✷✳✸✳✼✳ ❍➺ q✉↔✳ ●✐↔ sû (X, d) ❧➔ ❦❤æ♥❣ ❣✐❛♥ tü❛ ♠➯tr✐❝ ♥â♥ ✤➛② ✤õ ❍❛✉s❞♦r❢❢ ✈➔ f : X−→X ❧➔ ❤➔♠ ❧✐➯♥ tö❝ s❛♦ tỗ t {xn} tr X d(f xn , xn ) → ✈➔ d(xn , f xn ) → 0✳ ❑❤✐ ✤â✱ ♥➳✉ ϕ(x) = sup{d(f x, x), d(x, f x)}, x ∈ X ❧➔ ❤➔♠ tø X ✈➔♦ P t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ♥➯✉ tr♦♥❣ ỵ t f õ t ỡ ♥ú❛ ♥➳✉ ϕ ✈➔ d ❧✐➯♥ tö❝ t❤➻ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ f ❧➔ ❞✉② ♥❤➜t✳ ❈❤ù♥❣ ♠✐♥❤✳ ❚ø d(f xn, xn) → ✈➔ d(xn, f xn) → s✉② r❛ d(f xn, xn)+ d(xn , f xn ) → 0✳ ▼➦t ❦❤→❝✱ tø d(f xn , xn ) ≤ d(f xn , xn ) + d(xn , f xn ) ✈➔ d(xn , f xn ) ≤ d(f xn , xn ) + d(xn , f xn ), s✉② r❛ ✈ỵ✐ ♠å✐ n ≤ ϕ(xn ) ≤ d(f xn , xn ) + d(xn , f xn ), ∀n ❉♦ ✤â✱ t❤❡♦ ❍➺ q✉↔ ✶✳✸✳✼ t❛ ❝â (xn) ỵ tỗ t u ∈ X s❛♦ ❝❤♦ xn → u✳ ✣➳♥ ✤➙②✱ ❧➦♣ ỵ ữ tr ỵ t õ ự ỵ t r ỵ tr ổ tr ✤õ ❝õ❛ ❇❛♥❛❝❤ ✤÷đ❝ ♠ð rë♥❣ ❝❤♦ ❦❤ỉ♥❣ ❣✐❛♥ tü❛ ♠➯tr✐❝ ♥â♥ ✤➛② ✤õ X ✈➔ ❝→❝❤ ❝❤ù♥❣ ♠✐♥❤ ❝ô♥❣ t÷ì♥❣ tü ♥❤÷ tr÷í♥❣ ❤đ♣ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝✳ ❍➺ q✉↔ s❛✉ ✤➙② ❝❤♦ t❤➜② t❛ ❝â t❤➸ ♠ð rë♥❣ ◆❣✉②➯♥ ỵ ổ tỹ ♠➯tr✐❝ ♥â♥ ✈ỵ✐ ♠ët ❝→❝❤ ❝❤ù♥❣ ♠✐♥❤ ❦❤→❝✳ ❚r♦♥❣ ✣à♥❤ ♥❣❤➽❛ ✶✳✸✳✶✱ ♥➳✉ ❧➜② E ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❝→❝ sè t❤ü❝ R✱ P = [0, +∞) t❤➻ t❛ ✤÷đ❝ X ❧➔ ❦❤æ♥❣ ❣✐❛♥ tü❛ ♠➯tr✐❝✳ ✷✳✸✳✽✳ ❍➺ q✉↔✳◆➳✉ (X, d) ❧➔ ❦❤æ♥❣ ❣✐❛♥ tü❛ ♠➯tr✐❝ ♥â♥ ✤➛② ✤õ✱ ❍❛✉s✲ ❞♦r❢❢ ✈➔ f : X −→ X ❧➔ →♥❤ ①↕ ❝♦ t❤➻ f ❝â ❞✉② ♥❤➜t ♠ët ✤✐➸♠ ❜➜t ✤ë♥❣✳ ✸✵ ự f tỗ t↕✐ α ∈ [0, 1) s❛♦ ❝❤♦ d(f x, f y) ≤ αd(x, y), ∀x, y ∈ X ✭✶✳✽✮ ❉♦ ✤â t❛ ❝â (1 − α)d(x, y) ≤ d(x, y) − d(f x, f y) ≤ d(x, f x) + d(f y, y), ∀x, y ∈ X ✭✶✳✾✮ ❚❛ ①→❝ ✤à♥❤ ❤➔♠ ϕ : X −→ [0, +∞) ❜ð✐ ❝æ♥❣ t❤ù❝ ϕ(x) = sup{d(f x, x), d(f x, x)}, ∀x ∈ X ❚ø ✭✶✳✾✮ s✉② r❛ (1 − α)d(x, y) ≤ ϕ(x) + ϕ(y), ∀x, y ∈ X ❉♦ ✤â ✈ỵ✐ ♠é✐ c ∈ [0, +∞) ✈➔ ✈ỵ✐ x, y ∈ X ♠➔ c ≤ d(x, y) t❛ ❝â (1 − α)c ≤ ϕ(x) + ϕ(y) ❚ø ✤â s✉② r❛ r➡♥❣ ✈ỵ✐ ♠é✐ c ∈ (0, +∞) = int[0, +∞) = intP tỗ t à(c) := inf{(x) + (y) : c ≤ d(x, y)} ≥ (1 − α)c ❱➻ α ∈ [0, 1) ♥➯♥ (1 − α)c ∈ (0, +∞)✳ õ à(c) (0, +) ữ tọ ỵ t tứ s r❛ ✈ỵ✐ ♠é✐ x ∈ X t❛ ❝â d(f nx, f n+1x) → ✈➔ d(f n+1x, f nx) → ❦❤✐ n → ∞✳ ❚ø ✤â s✉② r❛ ♥➳✉ ❧➜② x ∈ X ✈➔ ✤➦t xn = f nx, n = 1, 2, t❤➻ t❛ ✤÷đ❝ ❞➣② {xn} tr♦♥❣ X t❤ä❛ ♠➣♥ d(xn , f xn ) → ✈➔ d(f xn , xn ) → 0✳ ❉♦ ✤â t❤❡♦ ❍➺ q✉↔ ✷✳✸✳✼✱ ❤➔♠ f ❝â ✤✐➸♠ ❜➜t ✤ë♥❣✳ ●✐↔ sû a ✈➔ b ❧➔ ✷ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ f ✳ ❑❤✐ ✤â✱ ♥➳✉ a = b t❤➻ tø α ∈ [0, 1) s✉② r❛ ≤ d(a, b) = d(f a, f b) ≤ αd(a, b) < d(a, b) ❚❛ ❝â ♠ët ✤✐➲✉ ♠➙✉ t❤✉➝♥✳ ❉♦ ✤â a = b✳ ❱➟② ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ f ❧➔ ❞✉② ♥❤➜t✳ ✸✶ ❑➌❚ ▲❯❾◆ ▲✉➟♥ ✈➠♥ ✤➣ ✤↕t ✤÷đ❝ ❝→❝ ❦➳t q✉↔ ❝❤➼♥❤ ♥❤÷ s❛✉✿ ✶✮ ❚r➻♥❤ ❜➔② ❦❤→✐ ♥✐➺♠ ✈➔ ♠ët sè t➼♥❤ ❝❤➜t ❝õ❛ ♥â♥ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✳ ✷✮ ✣÷❛ r❛ ✤à♥❤ ♥❣❤➽❛✱ ✈➼ ❞ư ✈➔ ♠ët sè t➼♥❤ ❝❤➜t tæ♣æ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ tü❛ ♠➯tr✐❝ ♥â♥ ✤÷đ❝ tr➻♥❤ ❜➔② tr♦♥❣ ♠ư❝ ✶✳✸✳ ✸✮ ✣÷❛ r ự ởt số ỵ sỹ tỗ t t →♥❤ ①↕ ❝♦ s✉② rë♥❣ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ tü❛ ♠➯tr✐❝ õ ú tữỡ tỹ ữ ố ợ ❦❤æ♥❣ ❣✐❛♥ tü❛ ♠➯tr✐❝ ✤➛② ✤õ ❤♦➦❝ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ õ õ ỵ q ỵ ỵ ữ r ự ởt số ỵ sỹ tỗ t↕✐ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ ❝→❝ →♥❤ ①↕ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ tü❛ ♠➯tr✐❝ ♥â♥ ✈ỵ✐ ♥â♥ ▼✐♥✐❤❡❞r❛❧ ♠↕♥❤✱ ✤â ❧➔ ỵ ỵ ỵ q✉↔ ✷✳✸✳✼✱ ❍➺ q✉↔ ✷✳✸✳✽✳ ✸✷ ❚⑨■ ▲■➏❯ ❚❍❆▼ ❑❍❷❖ ❬✶❪ ◆❣✉②➵♥ ❱➠♥ ❑❤✉➯✱ ▲➯ ▼➟✉ ❍↔✐ ✭✷✵✵✷✮✱ ❈ì sð ỵ tt t ◗✉è❝ ❣✐❛ ❍➔ ◆ë✐✳ ❬✷❪ ❏✳ ❑❡❧❧❡② ✭✶✾✼✸✮✱ ❚æ♣æ ✤↕✐ ữỡ ỗ ữớ ✭❞à❝❤✮✱ ◆①❜ ✣↕✐ ❤å❝ ✈➔ ❚r✉♥❣ ❤å❝ ❝❤✉②➯♥ ♥❣❤✐➺♣✱ ❍➔ ◆ë✐✳ ❬✸❪ ❆✳ ●r❛♥❛s ❛♥❞ ❏✳ ❉✉❣✉♥❞❥✐ ✭✷✵✵✽✮✱ ▼♦♥♦❣r❛♣❤s ✐♥ ▼❛t❤❡♠❛t✐❝s✳ ❋✐①❡❞ ♣♦✐♥t t❤❡♦r②✱ ❙♣r✐♥❣❡r ❬✹❪ ❍✉❛♥❣ ▲♦♥❣ ✲ ●✉❛♥❣ ❛♥❞ ❩❤❛♥❣ ❳✐❛♥ ✭✷✵✵✼✮✱ ❈♦♥❡ ♠❡tr✐❝ s♣❛❝❡ ❛♥❞ ❢✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠ ♦❢ ❝♦♥tr❛❝t✐✈❡ ♠❛♣♣✐♥❣s✱ ❏✳ ▼❛t❤✳ ❆♥❛❧✳ ❆♣♣❧✳ ✸✸✷✱ ♥♦✳ ✷✱ ✶✹✻✽ ✲ ✶✹✼✻✳ ❬✺❪ ❘✳ ❆✳ ❙t♦❧t❡♥❜❡r❣ ✭✶✾✻✾✮✱ ❖♥ q✉❛s✐ ✲ ♠❡tr✐❝ s♣❛❝❡s✱ ❉✉❦❡ ▼❛t❤✳ ❏✳✱ ✸✻✱ ✻✺✲✼✶✳ ❬✻❪ ❉✳ ❚✉r❦♦❣❧✉✱ ▼✳ ❆❜✉❧♦❤❛ ✭✷✵✶✵✮✱ ❈♦♥❡ ♠❡tr✐❝ s♣❛❝❡s ❛♥❞ ❢✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠s ✐♥ ❞✐❛♠❡tr✐❝❛❧❧② ❝♦♥tr❛❝t✐✈❡ ♠❛♣♣✐♥❣s✱ ❆❝t❛ ▼❛t❤❡♠❛t✐❝❛ s✐♥✐❝❛ ❊♥❣❧✐s❤ ❙❡r✐❡s ✷✻ ◆✉♠❜❡r ✭✸✮✱ ✹✽✾ ✲ ✹✾✻✳