▼Ö❈ ▲Ö❈ ❚r❛♥❣ ▼Ö❈ ▲Ö❈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ▼Ð ✣❺❯ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ữỡ ỹ tỗ t↕✐ ✤✐➸♠ ❜➜t ✤ë♥❣ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ o✲♠➯tr✐❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✶ ▼ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ởt số ỵ sỹ tỗ t t ①↕ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ o✲♠➯tr✐❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ữỡ ỹ tỗ t ✤✐➸♠ ❜➜t ✤ë♥❣ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ✈ỵ✐ ω ✲❦❤♦↔♥❣ ❝→❝❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✷✳✶ ▼ët số ỵ sỹ tỗ t t ❝õ❛ ❝→❝ →♥❤ ①↕ ✤ì♥ trà ✳ ✳ ✷✵ ✷✳✷ ỹ tỗ t t ✤❛ trà ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ❑➌❚ ▲❯❾◆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ❚⑨■ ▲■➏❯ ❚❍❆▼ ❑❍❷❖ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ é ỵ tt t ❧➔ ♠ët tr♦♥❣ ♥❤ú♥❣ ❝❤õ ✤➲ ✤÷đ❝ ♥❤✐➲✉ ❝❤✉②➯♥ ❣✐❛ ❣✐↔✐ t➼❝❤ q✉❛♥ t➙♠ ♥❣❤✐➯♥ ❝ù✉ ✈➔ t❤✉ ✤÷đ❝ ♥❤✐➲✉ ❦➳t q✉↔✳ ❑➳t q✉↔ q✉❛♥ trå♥❣ ✤➛✉ t✐➯♥ ♣❤↔✐ ❦➸ ỵ tr ổ ❣✐❛♥ ♠➯tr✐❝ ✤➛② ✤õ✳ ❉ü❛ ✈➔♦ ❦➳t q✉↔ ♥➔② ♥❣÷í✐ t❛ ✤➣ ♠ð rë♥❣ ♥â ❝❤♦ ♥❤✐➲✉ ❧♦↕✐ →♥❤ ①↕ ổ ởt tr ỳ ữợ rở õ ự sỹ tỗ t t ✤ë♥❣ ✈➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤✉♥❣ ❝õ❛ ❝→❝ →♥❤ ①↕ ✤ì♥ trà ✈➔ ❝→❝ →♥❤ ①↕ ✤❛ trà tr♦♥❣ ❦❤ỉ♥❣ tr ợ ỳ ữớ t ữủ t q q trồ ữợ rst ▼✳ ❆❛♠r✐ ❛♥❞ ❉✳ ❊■ ▼♦✉t❛✇❛❦✐❧✱ ❆❜❞✉❧ ▲❛t✐❢✱ ❲❛❢❛❛ ❆✳ ❆❧❜❛r✳✳✳ ▼ö❝ ✤➼❝❤ ❝õ❛ ❧✉➟♥ ✈➠♥ ❧➔ ❞ü❛ ✈➔♦ ❝→❝ t t ự ỵ tt t ự sỹ tỗ t t tr ổ otr sỹ tỗ t t ✤ë♥❣ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ✈ỵ✐ ω ✲❦❤♦↔♥❣ ❝→❝❤✳ ❱ỵ✐ ♠ư❝ ✤➼❝❤ ✤â✱ ❧✉➟♥ ✈➠♥ ✤÷đ❝ ❝❤✐❛ ❧➔♠ ❤❛✐ ❝❤÷ì♥❣ ữỡ ỹ tỗ t t tr ổ ❣✐❛♥ o✲♠➯tr✐❝ ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔②✱ ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ♠ët số t q sỹ tỗ t t ❜➜t ✤ë♥❣ ❝❤✉♥❣ ❝õ❛ ❝→❝ →♥❤ ①↕ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ otr ữỡ ỹ tỗ t t tr ổ tr ợ r ữỡ ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ❦❤→✐ ♥✐➺♠ ω✲❦❤♦↔♥❣ ❝→❝❤ tr♦♥❣ ❦❤ỉ♥❣ tr ởt số ỵ sỹ tỗ t↕✐ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ ❝→❝ →♥❤ ①↕ ✤ì♥ trà ✈➔ ✤❛ trà tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ✤➛② ✤õ ✈ỵ✐ ω✲❦❤♦↔♥❣ ❝→❝❤✳ ❈→❝ ❦➳t q✉↔ tr♦♥❣ ❧✉➟♥ ✈➠♥ ❝❤õ ②➳✉ ❧➔ ✤➣ ❝â tr♦♥❣ ❝→❝ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❝❤ó♥❣ tỉ✐ t➻♠ ❤✐➸✉ ✈➔ tr➻♥❤ ❜➔② t❤❡♦ ♠ư❝ ✤➼❝❤ ✤➣ ✤➦t r❛✳ ◆❣♦➔✐ ✈✐➺❝ ❤➺ t❤è♥❣✱ ❝❤ù♥❣ ♠✐♥❤ ❝❤✐ t✐➳t ♠ët sè ❦➳t q✉↔ ♠➔ tr♦♥❣ ❝→❝ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❝❤ù♥❣ ♠✐♥❤ ✈➢♥ t➢t ❤♦➦❝ ❜ä q✉❛ ❝❤ù♥❣ ♠✐♥❤ ❝❤ó♥❣ tỉ✐ ❝ơ♥❣ ✤÷❛ r❛ ✈➔ ❝❤ù♥❣ ♠✐♥❤ ✷ ♠ët sè ❦➳t q✉↔ ♠ỵ✐ ✤â ❧➔ ❱➼ ❞ư ✷✳✶✳✷ ✈➔ ỵ ữủ tỹ t rữớ ữợ sỹ ữợ t t ♥❣❤✐➯♠ ❦❤➢❝ ❝õ❛ P●❙✳❚❙✳ ✣✐♥❤ ❍✉② ❍♦➔♥❣✳ ❚→❝ ❣✐↔ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ ❝õ❛ ♠➻♥❤ ✤➳♥ ❚❤➛②✳ ❚→❝ ❣✐↔ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ❇❛♥ ❈❤õ ♥❤✐➺♠ ❑❤♦❛ ❙❛✉ ✤↕✐ ❤å❝✱ ❇❛♥ ❈❤õ ♥❤✐➺♠ ❑❤♦❛ ❚♦→♥ ✲ ❚r÷í♥❣ ✣↕✐ ❤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✉✳ ▼➦❝ ❞ò ✤➣ ❝â ♥❤✐➲✉ ❝è ❣➢♥❣ ♥❤÷ ❞♦ ❝á♥ ❤↕♥ ❝❤➳ ✈➲ ♠➦t ❦✐➳♥ t❤ù❝ ✈➔ t❤í✐ ❣✐❛♥ ♥➯♥ ❧✉➟♥ ✈➠♥ ổ tr ọ ỳ t sõt qỵ ổ õ õ ỵ ✈➠♥ ✤÷đ❝ ❤♦➔♥ t❤✐➺♥ ❤ì♥✳ ❱✐♥❤✱ t❤→♥❣ ✶✷ ♥➠♠ ✷✵✶✶ ❚→❝ ❣✐↔ ✸ ❈❍×❒◆● ✶ ❙Ü ❚➬◆ ❚❸■ ✣■➎▼ ❇❻❚ ✣❐◆● ❚❘❖◆● ❑❍➷◆● ●■❆◆ O✲▼➊❚❘■❈ ❈❤÷ì♥❣ ♥➔② tr➻♥❤ ❜➔② ♠ët số t q sỹ tỗ t t ❜➜t ✤ë♥❣ ❝❤✉♥❣ ❝õ❛ ❝→❝ →♥❤ ①↕ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ o✲♠➯tr✐❝✳ ✶✳✶✳ ▼❐❚ ❙➮ ❑■➌◆ ❚❍Ù❈ ❈❍❯❽◆ ❇➚ ▼ö❝ ♥➔② tr➻♥❤ ❜➔② ♠ët sè ❦❤→✐ ♥✐➺♠ ✈➔ ❦➳t q✉↔ ❝ì ❜↔♥ ❝➛♥ ❞ò♥❣ tr♦♥❣ ❧✉➟♥ ✈➠♥✳ ✶✳✶✳✶✳ ✣à♥❤ ♥❣❤➽❛✳ ❈❤♦ t➟♣ ❤đ♣ X ✈➔ ❤➔♠ d : X × X −→ R✳ ❍➔♠ d ✤÷đ❝ ❣å✐ ❧➔ ♠ët ♠➯tr✐❝ tr➯♥ X ♥➳✉ t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉✿ ✭✐✮ d(x, y) ≥ ✈➔ d(x, y) = ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ x = y❀ ✭✐✐✮ d(x, y) = d(y, x)❀ ✭✐✐✐✮ d(x, z) ≤ d(x, y) + d(y, z) ✈ỵ✐ ♠å✐ x, y, z ∈ X ✳ ❚➟♣ ❤đ♣ X ũ ợ ởt tr d tr õ ữủ ổ tr ỵ (X, d) ✤ì♥ ❣✐↔♥ ❤ì♥ ❧➔ X ✳ ✶✳✶✳✷✳ ✣à♥❤ ♥❣❤➽❛✳ ❈❤♦ t➟♣ ❤ñ♣ X ✳ ❍å τ ❝→❝ t➟♣ ❝♦♥ ❝õ❛ X ✤÷đ❝ ❣å✐ ❧➔ tỉ♣ỉ tr➯♥ X ♥➳✉ t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ ✭❚1✮ ∅, X ∈ τ ❀ ✭❚2✮ ◆➳✉ Gi ∈ τ, i ∈ I t❤➻ Gi ∈ τ ❀ i∈I ✭❚3✮ ◆➳✉ G1, G2 ∈ τ t❤➻ G1 ∩ G2 ∈ τ ✳ ❚➟♣ ❤đ♣ X ❝ị♥❣ ợ tổổ tr õ ữủ ổ tổổ ỵ (X, ) ỡ ❣✐↔♥ ❤ì♥ ❧➔ X ✳ ❈→❝ ♣❤➛♥ tû ❝õ❛ X ✤÷đ❝ ❣å✐ ❧➔ ✤✐➸♠ ❝õ❛ ❦❤ỉ♥❣ ❣✐❛♥ tỉ♣ỉ✳ ❈→❝ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ❧➔ ❦❤æ♥❣ ❣✐❛♥ tæ♣æ✱ tæ♣æ tr➯♥ ♥â ❣å✐ ❧➔ tæ♣æ s✐♥❤ ❜ð✐ ♠➯tr✐❝✳ ✹ ✶✳✶✳✸✳ ✣à♥❤ ♥❣❤➽❛✳ ❈❤♦ ❦❤æ♥❣ ❣✐❛♥ tæ♣æ X, A ⊂ X ✳ ❚➟♣ U ⊂ X ✤÷đ❝ ❣å✐ ❧➔ ❧➙♥ ❝➟♥ ❝õ❛ A ♥➳✉ ❝â t➟♣ ♠ð V tr♦♥❣ X s❛♦ ❝❤♦ A ⊂ V ⊂ U✳ ✶✳✶✳✹✳ ✣à♥❤ ♥❣❤➽❛✳ ❑❤æ♥❣ ❣✐❛♥ tæ♣æ X ✤÷đ❝ ❣å✐ ❧➔ T1✲❦❤ỉ♥❣ ❣✐❛♥ ♥➳✉ ❤❛✐ ✤✐➸♠ ❜➜t ❦ý x, y X, x = y tỗ t ❧➙♥ ❝➟♥ t÷ì♥❣ ù♥❣ Ux, Uy ❝õ❛ x ✈➔ y s❛♦ ❝❤♦ y ∈/ Ux ✈➔ x ∈/ Uy ✳ ✶✳✶✳✺✳ ✣à♥❤ ♥❣❤➽❛✳ ❉➣② xn tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ tæ♣æ X ữủ tử tợ x X ợ ộ U x tỗ t n0 ∈ N s❛♦ ❝❤♦ xn ∈ U ✈ỵ✐ ♠å✐ n ≥ n0 ❑❤✐ ✤â t❛ ✈✐➳t xn → x✳ ✶✳✶✳✻✳ ✣à♥❤ ♥❣❤➽❛✳ ❑❤ỉ♥❣ ❣✐❛♥ tỉ♣ỉ X ✤÷đ❝ ❣å✐ ❧➔ T2✲❦❤ỉ♥❣ ❣✐❛♥ ❤❛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 (X, d) (Y, ) ❧➔ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ✈➔ →♥❤ ①↕ f : X −→ Y ✳ ❑❤✐ ✤â ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ t÷ì♥❣ ✤÷ì♥❣ ✭✶✮ f ❧✐➯♥ tư❝ t↕✐ x ∈ X > tỗ t > s❛♦ ❝❤♦ y ∈ X, d(x, y) < δ t❤➻ ρ(f (x), f (y)) < ε; ✭✸✮ ▼å✐ ❞➣② {xn} ⊂ X s❛♦ ❝❤♦ xn → x t❤➻ f (xn) → ✺ f (x)✳ ✶✳✶✳✾✳ ✣à♥❤ ♥❣❤➽❛✳ ●✐↔ sû X ❧➔ ❦❤æ♥❣ ❣✐❛♥ tæ♣æ✱ ❤➔♠ f : X −→ R ✤÷đ❝ ❣å✐ ❧➔ ♥û❛ ❧✐➯♥ tư❝ tr➯♥ t↕✐ x0 ∈ X ♥➳✉ lim sup f (x) ≤ f (x0 ) x→ x0 ❍➔♠ f ✤÷đ❝ ❣å✐ ❧➔ ♥û❛ ❧✐➯♥ tö❝ tr➯♥ tr➯♥ X ♥➳✉ ♥â ❧✐➯♥ tö❝ tr➯♥ t↕✐ ♠å✐ x ∈ X✳ ❍➔♠ f ✤÷đ❝ ❣å✐ ❧➔ ♥û❛ tử ữợ (f ) ỷ tử tr➯♥✱ tr♦♥❣ ✤â (−f )(x) = −f (x) ✈ỵ✐ ♠å✐ x ∈ X ◆â✐ ❝→❝❤ ❦❤→❝✱ ❤➔♠ f ✤÷đ❝ ❣å✐ ỷ tử ữợ t x0 X lim inf f (x) ≥ f (x0 ) x→ x0 ✣æ✐ ❦❤✐✱ t❛ ✈✐➳t lim inf f (x)✳ x→x lim f (x), x→x0 lim f (x) x→x0 ❧➛♥ ❧÷đt t❤❛② ❝❤♦ lim sup f (x) x→x0 ✈➔ ✶✳✶✳✶✵✳ ✣à♥❤ ỵ sỷ X ổ tổổ f : X −→ R✳ ❑❤✐ ✤â✱ ❢ ♥û❛ ❧✐➯♥ tö❝ tr ỷ tử ữợ tữỡ ự ❦❤✐ ✈ỵ✐ ♠å✐ r ∈ R✱ t➟♣ {x ∈ X : f (x) < r} ({x ∈ X : f (x) > r}✱ t÷ì♥❣ ù♥❣✮ ♠ð tr♦♥❣ ❳✳ ✶✳✶✳✶✶✳ ✣à♥❤ ỵ sỷ ổ tổổ f : X −→ R✳ ❑❤✐ ✤â✱ ❢ ❧✐➯♥ tö❝ t↕✐ x ∈ X ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ f ❧✐➯♥ tö❝ tr tử ữợ t ❈❤♦ X ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝✳ ▼ët ❞➣② {xn} tr♦♥❣ ❣å✐ ❧➔ ❞➣② ❈❛✉❝❤② ♥➳✉ ✈ỵ✐ ♠å✐ ε > 0, tỗ t n0 N s ợ n ✈➔ m ≥ n0 t❤➻ d(xn , xm ) < ε✳ ▼å✐ ❞➣② ❤ë✐ tư ❧➔ ❞➣② ❈❛✉❝❤②✳ ❑❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ X ❣å✐ ❧➔ ✤➛② ✤õ ♥➳✉ ♠å✐ ❞➣② ❈❛✉❝❤② tr♦♥❣ X ✤➲✉ ❤ë✐ tö✳ ❚➟♣ ❝♦♥ A ⊂ X ❣å✐ ❧➔ t➟♣ ✤➛② ✤õ ♥➳✉ ♥â ✤➛② ✤õ ✈ỵ✐ ♠➯tr✐❝ ❝↔♠ s✐♥❤✳ ▼å✐ t➟♣ ❝♦♥ ✤➛② ✤õ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ❧➔ t➟♣ ✤â♥❣✱ ♠å✐ t➟♣ ❝♦♥ ✤â♥❣ ❝õ❛ ♠ët ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ✤➛② ✤õ ❧➔ t➟♣ ✤➛② ✤õ✳ X ✻ ✶✳✶✳✶✸✳ ✣à♥❤ ♥❣❤➽❛✳ ●✐↔ sû X, Y ❧➔ t rộ ỵ 2Y t➜t ❝↔ ❝→❝ t➟♣ ❝♦♥ ❝õ❛ Y ✳ ❚❛ ❣å✐ ♠é✐ →♥❤ ①↕ tø X ✈➔♦ Y ❧➔ ♠ët →♥❤ ①↕ ✤ì♥ trà ❤❛② ❤➔♠ ✤ì♥ trà ✈➔ ❣å✐ ♠é✐ →♥❤ ①↕ tø X ✈➔♦ ♠ët ❤å ❝♦♥ ❝õ❛ 2Y ❧➔ ♠ët →♥❤ ①↕ ✤❛ trà ❤❛② ❤➔♠ ✤❛ trà✳ P (Y ) ✶✳✶✳✶✹✳ ✣à♥❤ ♥❣❤➽❛✳ ●✐↔ sû f : X−→X ✈➔ T : X−→U ✈ỵ✐ U ⊂ 2X ✳ ✣✐➸♠ x ∈ X ✤÷đ❝ ❣å✐ ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ f ♥➳✉ f (x) = x✳ ✣✐➸♠ x ∈ X ✤÷đ❝ ❣å✐ ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ T ♥➳✉ x ∈ T (x)✳ ✣✐➸♠ x ∈ X ✤÷đ❝ ❣å✐ ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤✉♥❣ ❝õ❛ f ✈➔ T ♥➳✉ x = f (x) ∈ T (x)✳ ✶✳✶✳✶✺✳ ✣à♥❤ ♥❣❤➽❛✳ ●✐↔ sû X ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ tæ♣æ✳ ❍➔♠ d : X ì ữủ ởt otr tr X ♥➳✉ t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉✿ ✭✶✮ d(x, y) ≥ ✈ỵ✐ ♠å✐ x, y ∈ X ❀ ✭✷✮ d(x, y) = ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ x = y❀ ✭✸✮ ❚➟♣ ❝♦♥ U ⊂ X ❧➔ ♠ð ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ d(x, X\U ) > ✈ỵ✐ ♠å✐ x ∈ U tr♦♥❣ ✤â d(x, X\U ) = ✐♥❢{d(x, y) : y ∈ X\U } ❍➔♠ d ✤÷đ❝ ❣å✐ ❧➔ ♠ët o✲♠➯tr✐❝ ♠↕♥❤ ♥➳✉ d ❧➔ o✲♠➯tr✐❝ ✈➔ ✈ỵ✐ ♠é✐ x ∈ X ✱ ✈ỵ✐ ♠é✐ r > ❤➻♥❤ ❝➛✉ X−→ R B(x, r) = {y ∈ X : d(x, y) < r} ❧➔ ♠ët ❧➙♥ ❝➟♥ ❝õ❛ x✳ ❑❤ỉ♥❣ ❣✐❛♥ tỉ♣ỉ X ❝ị♥❣ ✈ỵ✐ ♠ët o✲♠➯tr✐❝ d tr õ ữủ ổ otr ỵ ❤✐➺✉ ❧➔ (X, d) ❤♦➦❝ X ♥➳✉ ❦❤æ♥❣ ❝➛♥ ❝❤➾ r❛ d✳ ●✐↔ sû d ❧➔ ♠ët o✲♠➯tr✐❝ tr➯♥ X ✳ ✣➦t τd = {U ⊂ X : ∀x ∈ U, ∃B(x, ε) ⊂ U } ✶✳✶✳✶✻✳ ▼➺♥❤ ✤➲✳ ✶✮ ❚➟♣ ❝♦♥ U ❧➔ ♠ð ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ U ∈ τd✳ ◆â✐ ❝→❝❤ ❦❤→❝ τd trị♥❣ ✈ỵ✐ tỉ♣ỉ ❝õ❛ X ✳ ✼ ✷✮ ◆➳✉ {xn} ⊂ X ✈➔ x ∈ X s❛♦ ❝❤♦ d(x, xn) → t❤➻ xn → x✳ ❈❤ù♥❣ ♠✐♥❤✳ ✶✮ ●✐↔ sû U ❧➔ t➟♣ ♠ð tr♦♥❣ X ✈➔ x ∈ U ✳ ❑❤✐ ✤â✱ t otr t tỗ t r > s❛♦ ❝❤♦ d(x, X\U ) = r✳ ❚ø ✤â s✉② r❛ r B x, ⊂ U✳ ❚❤➟t ✈➟②✱ ✈ỵ✐ ♠é✐ y ∈ B x, 2r ⊂ U t❛ ❝â d(x, y) < 2r ✳ ❉♦ ✤â y ∈/ X\U ✱ tù❝ ❧➔ y ∈ U ✳ ◆❤÷ ✈➟② U ∈ τd✳ ◆❣÷đ❝ ❧↕✐✱ ❣✐↔ sû U ∈ τd x U õ tỗ t > s❛♦ ❝❤♦ B(x, ε) ⊂ U ✳ ❱ỵ✐ y ∈ X\U t❛ ❝â x ∈/ U ✳ ❉♦ ✤â d(x, y) ≥ ε✳ ❚ø ✤â s✉② r❛ d(x, X\U ) ≥ ε > ❚❤❡♦ ✣à♥❤ ♥❣❤➽❛ ❝õ❛ o✲♠➯tr✐❝ t❤➻ U ❧➔ t➟♣ ♠ð tr♦♥❣ X ✳ ✷✮ ●✐↔ sû {xn} ⊂ U, x ∈ X s❛♦ ❝❤♦ d(x, xn) → 0✳ ❱ỵ✐ ❜➜t ❦ý ❧➙♥ ❝➟♥ U x t tỗ t r > s B(x, r) ⊂ U ✳ ❱➻ d(x, xn) → tỗ t số tỹ n0 s d(x, xn) < r ✈ỵ✐ ♠å✐ n ≥ n0✳ ❉♦ ✤â xn ∈ B(x, r) ⊂ U ✈ỵ✐ ♠å✐ n ≥ n0✳ ❱➟② xn → x d ✶✳✶✳✶✼✳ ▼➺♥❤ ✤➲✳ ◆➳✉ (X, d) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♦✲♠➯tr✐❝ ❍❛✉s❞♦r❢❢ t❤➻ xn −→ x ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ xn → x✳ ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû xn → x✳ ❑❤✐ ✤â✱ ✈➻ X ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ❍❛✉s❞♦r❢❢ ♥➯♥ ✈ỵ✐ ♠é✐ ε > 0, B(x, ) x õ tỗ t↕✐ n0 ∈ N s❛♦ ❝❤♦ xn ∈ B(x, ε) ✈ỵ✐ ♠å✐ n ≥ n0 ✱ tù❝ ❧➔ ≤ d(x, xn ) < ε ✈ỵ✐ ♠å✐ n ≥ n0 ❚ø ✤â s✉② r❛ d(x, xn) → ✶✳✶✳✶✽✳ ✣à♥❤ ♥❣❤➽❛✳ ●✐↔ sû (X, d) ❧➔ ❦❤æ♥❣ ❣✐❛♥ o✲♠➯tr✐❝ ✈➔ {xn} ❧➔ ♠ët ❞➣② tr♦♥❣ X ✳ ❉➣② {xn} ✤÷đ❝ ❣å✐ ❧➔ ❞➣② ❈❛✉❝❤② ♥➳✉ ✈ỵ✐ ♠å✐ ε > tỗ t n0 N s ợ n ≥ n0 ✈➔ ✈ỵ✐ ♠å✐ m ∈ N t❛ ❝â d(xn , xn+m ) < ε✳ ✽ ❑❤æ♥❣ ❣✐❛♥ X ✤÷đ❝ ❣å✐ ❧➔ ✤➛② ✤õ ♥➳✉ {xn} ❧➔ ♠ët ❞➣② tr X t tỗ t x X s ❝❤♦ {xn} ❤ë✐ tư tỵ✐ x✳ ✶✳✶✳✶✾✳ ▼➺♥❤ ✤➲✳ ❱ỵ✐ ❦❤æ♥❣ ❣✐❛♥ ♦✲♠➯tr✐❝ (X, d) ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ ❧➔ t÷ì♥❣ ✤÷ì♥❣ ✭❛✮ ▼å✐ ❞➣② d✲❤ë✐ tư ❧➔ ❞➣② ❈❛✉❝❤②✳ d d ✭❜✮ ◆➳✉ {xn}, {yn} ❧➔ ❝→❝ ❞➣② tr♦♥❣ X ✈➔ x ∈ X s❛♦ ❝❤♦ xn −→ x, yn −→ x t❤➻ n→∞ lim d(xn , yn ) = ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû ✤✐➲✉ ❦✐➺♥ ✭❛✮ ✤÷đ❝ t❤ä❛ ♠➣♥ ✈➔ {xn}, {yn} ❧➔ ❤❛✐ ❞➣② d d tr♦♥❣ X s❛♦ ❝❤♦ xn −→ x, yn −→ x ✈➔ x ∈ X ✳ ❱ỵ✐ ♠é✐ n = 1, 2, ✤➦t z2n−1 = x, z2n = yn ❑❤✐ ✤â✱ tø d(x, xn) → ✈➔ d(x, yn) → s✉② r❛ n→∞ lim d(x, zn ) → 0✱ tù❝ ❧➔ d zn −→ x✳ ❚❤❡♦ ✤✐➲✉ ❦✐➺♥ ✭❛✮✱ {zn } ❧➔ ❞➣② ❈❛✉❝❤②✳ ❉♦ ✤â✱ ợ > tỗ t n0 N s❛♦ ❝❤♦ d(zn , zn+m ) < ε ✈ỵ✐ ♠å✐ n ≥ n0 , m ∈ N ❚ø ✤â s✉② r❛ d(xn , yn ) = d(z2n−1 , z2n ) < ε ✈ỵ✐ ♠å✐ n ≥ n0 ❉♦ ✤â n→∞ lim d(xn , yn ) = 0✳ ◆❣÷đ❝ ❧↕✐✱ ❣✐↔ sû ✤✐➲✉ ❦✐➺♥ ✭❜✮ ✤÷đ❝ t❤ä❛ ♠➣♥ ✈➔ {xn} ❧➔ ❞➣② d✲❤ë✐ tư tỵ✐ x ∈ X ✳ ❚❛ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ {xn } ❧➔ ❞➣② ❈❛✉❝❤②✳ ●✐↔ sû {xn } ổ õ tỗ t ε > s❛♦ ❝❤♦ ✈ỵ✐ ♠é✐ n ∈ N tỗ t mn > n kn N tọ ♠➣♥ d(xm , xm +k ) > ε ✭✯✮ ❑❤æ♥❣ ♠➜t t➼♥❤ tê♥❣ q✉→t ❝â t❤➸ ❣✐↔ t❤✐➳t mn + kn < mn+1 ✈ỵ✐ ♠å✐ n✳ ❑❤✐ ✤â {xm } ✈➔ {xm +k } ❧➔ ❞➣② ❝♦♥ ❝õ❛ ❞➣② {xn}✳ ❚ø d(x, xn) → s✉② r❛ n n n n n n lim d(x, xmn ) = lim d(x, xmn +kn ) = n→∞ n→∞ ✾ ❚❤❡♦ ✤✐➲✉ ❦✐➺♥ ✭❜✮ t❛ ❝â lim d(xmn , xmn +kn ) = n→∞ ✣✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥ ✈ỵ✐ (∗)✳ ❉♦ ✤â {xn} ❧➔ ❞➣② ❈❛✉❝❤②✳ ✶✳✷✳ ▼❐❚ ❙➮ ✣➚◆❍ ▲Þ ❱➋ ❙Ü ❚➬◆ ❚❸■ ✣■➎▼ ❇❻❚ ✣❐◆● ❈Õ❆ ❈⑩❈ ⑩◆❍ ❳❸ ❚❘❖◆● ❑❍➷◆● ●■❆◆ o✲▼➊❚❘■❈ ✶✳✷✳✶✳ ✣à♥❤ ♥❣❤➽❛✳ ●✐↔ sû X ❧➔ ❦❤æ♥❣ ❣✐❛♥ o✲♠➯tr✐❝✳ ❚❛ ♥â✐ X ❜à ❝❤➦♥ ♥➳✉ d(X) = sup{d(x, y) : x, y ∈ X} < ∞ ✶✳✷✳✷✳ ✣à♥❤ ♥❣❤➽❛✳ ●✐↔ sû f : X−→ X ✳ ✣✐➸♠ a ∈ X ✤÷đ❝ ❣å✐ ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ f ♥➳✉ f (a) = a✳ ❚❛ ✤➣ ❜✐➳t r➡♥❣ ♠é✐ →♥❤ ①↕ ❝♦ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ✤➛② ✤õ ❝â ✤✐➸♠ ❜➜t ✤ë♥❣ ❞✉② ♥❤➜t✳ ✣à♥❤ ỵ s t ụ ú ổ otr ỵ (X, d) ❧➔ ❦❤æ♥❣ ❣✐❛♥ o✲♠➯tr✐❝ ❍❛✉s❞♦r❢❢ ✤➛② ✤õ s❛♦ ❝❤♦ d(f (X)) = sup{d(x, y) : x, y ∈ f (X)} < ∞ t❤➻ ♠é✐ →♥❤ ①↕ ❝♦ f : X−→ X ❝â ✤✐➸♠ ❜➜t ✤ë♥❣ ❞✉② ♥❤➜t✳ ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû f : X−→ X ❧➔ →♥❤ ①↕ ❝♦✱ tỗ t [0, 1) s d(f (x), f (y)) ≤ αd(x, y), ✈ỵ✐ ♠å✐ x, y ∈ X ▲➜② x1 ∈ X ✈➔ ✤➦t x2 = f (x1)✳ ◆➳✉ d(x1, x2) = t❤➻ x1 = x2 = f (x1)✱ ❞♦ ✤â x1 ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ f ✳ ●✐↔ sû d(x1 , x2 ) > 0✳ ❚❛ ①➙② ❞ü♥❣ ❞➣② {xn } ♥❤÷ s❛✉ xn+1 = f (xn ); ✈ỵ✐ n = 1, 2, ✶✵ d(Au, Bv) ≤ ϕ(max{d(Su, T u), d(Su, Bv), d(T v, Bv)}) ≤ ϕ(max{d(T v, Bv), d(Au, Bv)}) ≤ ϕ(d(Au, Bv)) < d(Au, Bv) ✣✐➲✉ ♥➔② ổ ỵ õ Au = Su = T v = Bv✳ ❚❛ ❧↕✐ ❝â B ✈➔ T t÷ì♥❣ t❤➼❝❤ ②➳✉ ♥➯♥ BT v = T Bv ✈➔ ❞♦ ✤â T T v = T Bv = BT v = BBv ❇➙② ❣✐í t❛ ❝❤ù♥❣ ♠✐♥❤ Au ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ A, B, T ✈➔ S ✳ ●✐↔ sû AAu = Au✳ ❑❤✐ ✤â d(AAu, Au) = d(AAu, Bv) ≤ ϕ(max{d(SAu, T v), d(SAu, Bv), d(T v, Bv)}) ≤ ϕ(max{d(AAu, Au), d(AAu, Au), d(Au, Au)}) ≤ ϕ(d(AAu, Au)) < d(AAu, Au) ổ ỵ õ Au = AAu = SAu ✈➔ Au ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ ❝↔ A ✈➔ S ✳ ❍♦➔♥ t♦➔♥ t÷ì♥❣ tü t❛ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ Au ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ B ✈➔ T ✳ ❱➟② Au ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ A, B, T ✈➔ S ✳ ❈✉è✐ ❝ò♥❣✱ ❣✐↔ sû tỗ t u, v X s Au = Bu = T u = Su = u ✈➔ Av = Bv = T v = Sv = v ♥❤÷♥❣ u = v ✳ ❑❤✐ ✤â✱ tø ✭✶✮ t❛ ❝â d(u, v) = d(Au, Bv) ≤ ϕ(max{d(Su, T v), d(Su, Bv), d(T v, Bv)}) ≤ ϕ(d(u, v)) < d(u, v) ổ ỵ Au t ❝❤✉♥❣ ❞✉② ♥❤➜t ❝õ❛ A, B, T ✈➔ S ✶✾ ❈❍×❒◆● ✷ ❙Ü ❚➬◆ ❚❸■ ✣■➎▼ ❇❻❚ ✣❐◆● ❚❘❖◆● ❑❍➷◆● ●■❆◆ ▼➊❚❘■❈ ❱❰■ ω✲❑❍❖❷◆● ❈⑩❈❍ ❈❤÷ì♥❣ ♥➔② tr➻♥❤ ❜➔② ❦❤→✐ ♥✐➺♠ ω✲❦❤♦↔♥❣ ❝→❝❤ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ✈➔ ♠ët sè ỵ sỹ tỗ t t ❝→❝ →♥❤ ①↕ ✤ì♥ trà ✈➔ ✤❛ trà tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ✤➛② ✤õ ✈ỵ✐ ω✲❦❤♦↔♥❣ ❝→❝❤✳ ✷✳✶✳ ▼❐❚ ❙➮ ✣➚◆❍ ▲Þ ❱➋ ❙Ü ❚➬◆ ❚❸■ ✣■➎▼ ❇❻❚ ✣❐◆● ❈Õ❆ ❈⑩❈ ⑩◆❍ ❳❸ ✣❒◆ ❚❘➚ ✷✳✶✳✶✳ ✣à♥❤ ♥❣❤➽❛✳ ●✐↔ sû (X, d) ❧➔ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝✳ ❍➔♠ sè ✤÷đ❝ ❣å✐ ❧➔ ω✲❦❤♦↔♥❣ ❝→❝❤ tr➯♥ X ♥➳✉ ♥â t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ ✈ỵ✐ ❜➜t ❦ý x, y, z ∈ X (ω1 ) p(x, z) ≤ p(x, y) + p(y, z)❀ (ω2 ) →♥❤ ①↕ p(x, ) : X−→ [0, ) ỷ tử ữợ (3 ) ợ t ý > tỗ t > s❛♦ ❝❤♦ p(z, x) ≤ δ ✈➔ p(z, y) ≤ t d(x, y) ú ỵ r ♥â✐ ❝❤✉♥❣ p(x, y) = p(y, x) ✈ỵ✐ x, y ∈ X ✳ p : X × X−→ [0, ∞) ✷✳✶✳✷✳ ❱➼ ❞ư✳ ✶✮ ◆➳✉ (X, d) ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ t❤➻ d ❝ô♥❣ ❧➔ ♠ët ω✲ ❦❤♦↔♥❣ ❝→❝❤ tr➯♥ X ✳ ❚❤➟t ✈➟②✱ ❤✐➸♥ ♥❤✐➯♥ d t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ (ω1), (ω2)✳ ❱ỵ✐ ♠å✐ ε > ❧➜② δ = 12 ε✳ ❑❤✐ ✤â✱ ✈ỵ✐ ♠å✐ x, y, z ∈ X ♠➔ d(z, x) < δ ✈➔ d(z, y) < δ t❤➻ x ✈➔ y t❤✉ë❝ B(z, δ)✱ tr õ B(z, ) ỵ t z✱ ❜→♥ ❦➼♥❤ δ tr♦♥❣ X ✳ ❚ø x✱ y ∈ B(z, δ) s✉② r❛ d(x, y) < 2δ = ε✳ ❉♦ ✤â d t❤ä❛ ♠➣♥ (ω3 )✳ ✷✮ ●✐↔ sû d ❧➔ ♠➯tr✐❝ t❤ỉ♥❣ t❤÷í♥❣ tr➯♥ ❦❤ỉ♥❣ ❣✐❛♥ ❝→❝ sè t❤ü❝ R ✈➔ p : R × R−→ [0, ∞) ❧➔ ❤➔♠ ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ p(x, y) = |y| ợ (x, y) R ì R ❍✐➸♥ ♥❤✐➯♥ p t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ (ω1)✳ ❚ø →♥❤ ①↕ t −→ |t| ❧➔ ❧✐➯♥ tö❝ tr➯♥ R s✉② r❛ ✈ỵ✐ ♠é✐ x ∈ R✱ ❤➔♠ p(x, y) = |y| ✈ỵ✐ ♠å✐ y ∈ R ❧➔ ❧✐➯♥ tư❝ tr➯♥ R✳ ❉♦ ✤â p t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ (ω2 )✳ ❱ỵ✐ ♠å✐ ε > 0✱ ❧➜② δ = 12 ε✳ ❑❤✐ ✤â✱ ✈ỵ✐ ♠å✐ x, y, z ∈ R ♠➔ p(z, x) = |x| < δ, p(z, y) = |y| < δ t❛ ❝â d(x, y) = |x − y| < |x| + |y| < 2δ = ε ❉♦ ✤â p t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ (ω3)✳ ❱➟② p ❧➔ ω✲❦❤♦↔♥❣ ❝→❝❤ tr➯♥ R✳ ✷✳✶✳✸✳ ❇ê ✤➲ ✭❬✺❪✮✳ ❈❤♦ X ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ✈ỵ✐ ♠➯tr✐❝ ❞✱ ✈➔ p ❧➔ ♠ët ω ✲❦❤♦↔♥❣ ❝→❝❤ tr➯♥ X ✳ ●✐↔ sû {xn } ✈➔ {yn } ❧➔ ❝→❝ ❞➣② tr♦♥❣ X ✱ {αn } ✈➔ {βn } ❧➔ ❝→❝ ❞➣② tr♦♥❣ R+ ❤ë✐ tư tỵ✐ 0✱ ✈➔ x, y, z ∈ X ✳ ❑❤✐ ✤â✱ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ ✤÷đ❝ t❤ä❛ ♠➣♥✿ ✭✐✮ ◆➳✉ p(xn, y) ≤ αn ✈➔ p(xn, z) ≤ βn ✈ỵ✐ ♠å✐ n ∈ N t❤➻ y = z ✳ ✣➦❝ ❜✐➺t ♥➳✉ p(x, y) = ✈➔ p(x, z) = t❤➻ y = z ❀ ✭✐✐✮ ◆➳✉ p(xn, yn) ≤ αn ✈➔ p(xn, z) ≤ βn ✈ỵ✐ ♠å✐ n ∈ N t❤➻ {yn} ❤ë✐ tö ✈➲ z❀ ✭✐✐✐✮ ◆➳✉ p(xn, xm) ≤ αn ✈ỵ✐ ♠å✐ n, m ∈ N ♠➔ m > n t❤➻ {xn} ❧➔ ❞➣② ❈❛✉❝❤②❀ ✭✐✈✮ ◆➳✉ p(y, xn) ≤ αn ✈ỵ✐ ♠å✐ n ∈ N t {xn} ỵ (X, d) ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ✤➛② ✤õ ✈ỵ✐ ω✲❦❤♦↔♥❣ ❝→❝❤ ♣✳ ●✐↔ sû f, g : X−→ X ✈➔ ϕ : X−→ [0, ∞) t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥✿ g(X) f (X) ỗ t t X s❛♦ ❝❤♦ p(t, gx) ≤ r.p(t, f x) + [ϕ(f x) − ϕ(gx)] ✈ỵ✐ ♠å✐ x ∈ X ✈➔ r ♥➔♦ ✤â t❤✉ë❝ [0, 1)❀ ✷✶ ✭✐✐✐✮ ◆➳✉ {xn}n∈N tr♦♥❣ X t❤ä❛ ♠➣♥ lim p(t, f xn ) = lim p(t, gxn ) = n→∞ n→∞ t❤➻ lim max{p(t, f xn ), p(t, gxn ), p(f gxn , gf xn )} = 0; n→∞ ✭✐✈✮ ◆➳✉ u ∈ X ♠➔ u = f u ❤♦➦❝ u = gu t❤➻ ✐♥❢{p(u, f x) + p(u, gx) + p(f gx, gf x) : x ∈ X} > ❑❤✐ ✤â✱ f ✈➔ g ❝â ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤✉♥❣ ❞✉② ♥❤➜t tr♦♥❣ X ✳ ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû x0 ∈ X ứ s r tỗ t {xn} ∈ X s❛♦ ❝❤♦ gxn−1 = f xn ✈ỵ✐ ♠å✐ n ≥ 1✳ ❚ø ✤✐➲✉ ❦✐➺♥ ✭✐✐✮ t❛ ❝â p(t, f xj+1 ) = p(t, gxj ) ≤ r.p(t, f xj ) + [ϕ(f xj ) − ϕ(gxj )] ❉♦ ✤â n−1 n−1 p(t, f xj+1 ) ≤ r j=0 n−1 [ϕ(f xj ) − ϕ(gxj )] p(t, f xj ) + j=0 j=0 ❙✉② r❛ n p(t, f xj ) ≤ j=0 ≤ ✣✐➲✉ ♥➔② ❝❤ù♥❣ tä r r p(t, f x0 ) + [ϕ(f x0 ) − ϕ(f xn )] 1−r 1−r r p(t, f x0 ) + ϕ(f x0 ) 1−r 1−r ∞ p(t, f xn ) n=1 ❤ë✐ tö✳ ❱➻ ✈➟② lim p(t, f xn ) = lim p(t, gxn ) = n→∞ n→∞ ●✐↔ sû r➡♥❣ t = f t ❤♦➦❝ t = gt✳ ❚ø ✤✐➲✉ ❦✐➺♥ ✭✐✐✐✮ ✈➔ ✭✐✈✮ t❛ ❝â < ✐♥❢{p(t, f x) + p(t, gx) + p(f gx, gf x) : x ∈ X} ≤ ✐♥❢{p(t, f xn ) + p(t, gxn ) + p(f gxn , gf xn ) : n ∈ N} = ✷✷ ✣➙② ❧➔ ♠ët ✤✐➲✉ ♠➙✉ t❤✉➝♥✳ ❉♦ ✤â t = f t = gt✳ ❱➟② t ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤✉♥❣ ❝õ❛ f ✈➔ g✳ ❚❛ ❝❤ù♥❣ ♠✐♥❤ t ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤✉♥❣ ❞✉② ♥❤➜t ❝õ❛ f ✈➔ g✳ ●✐↔ sû u ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤✉♥❣ ❝õ❛ f ✈➔ g✳ ❚❤❡♦ ✤✐➲✉ ❦✐➺♥ ✭✐✐✮✱ p(t, t) = p(t, gt) ≤ r.p(t, f t) + [ϕ(f t) − ϕ(gt)] = r.p(t, t), p(t, u) = p(t, gu) ≤ r.p(t, f u) + [ϕ(f u) − ϕ(gu)] = r.p(t, u) ❱➻ ✈➟② p(t, t) = p(t, u) = 0✳ ❚ø ❇ê ✤➲ ✷✳✶✳✸ t❛ ✤÷đ❝ t = u✳ ❉♦ ✤â t ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤✉♥❣ ❞✉② ♥❤➜t ❝õ❛ f ✈➔ g ✷✳✶✳✺✳ ◆❤➟♥ ①➨t✳ ❈❤♦ f ❧➔ →♥❤ ①↕ ❧✐➯♥ tư❝ tø ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ (X, d) ✈➔♦ ❝❤➼♥❤ ♥â ✈➔ f ❝â ✤✐➸♠ ❜➜t ✤ë♥❣ tr♦♥❣ X ✳ ❑❤✐ õ tỗ t p t X, g : X−→ X ✈➔ ϕ : X−→ [0, ∞) t❤ä❛ tr ỵ ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû z ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ f ✳ ❑❤✐ ✤â✱ ❧➜② t = z ✈➔ ①→❝ ✤à♥❤ g : X−→ X, ϕ : X−→ [0, ∞) ❜ð✐ ❝→❝ ❝æ♥❣ t❤ù❝ gx = z, ϕ(x) = ✈ỵ✐ ♠å✐ x ∈ X ❚❛ ①→❝ ✤à♥❤ p : X ì X [0, ) ợ p(x, y) = max{d(f x, x), d(f x, y), d(f x, f y)} ∀x, y ∈ X ●✐↔ t❤✐➳t r➡♥❣ lim p(t, f xn ) = lim p(t, gxn ) = n→∞ n→∞ ❑❤✐ ✤â✱ t❛ ❦✐➸♠ tr❛ ✤÷đ❝ ❝→❝ ✤✐➲✉ ❦✐➺♥ ✭✐✮✱ tr ỵ ữủ tọ ỵ sỷ (X, d) ổ ❣✐❛♥ ♠➯tr✐❝ ✤➛② ✤õ✱ ❢ ✈➔ g : X−→ X ❧➔ ❝→❝ →♥❤ ①↕ ❧✐➯♥ tư❝ ✈➔ t÷ì♥❣ t❤➼❝❤ s❛♦ g(X) f (X) õ tỗ t ❤➔♠ ❧✐➯♥ tö❝ ϕ : X−→ [0, ∞), t ∈ X ✈➔ r ∈ [0, 1) s❛♦ ❝❤♦ d(t, gx) ≤ rd(t, f x) + ϕ(f x) − ϕ(gx) ✷✸ ✭✶✮ ✈ỵ✐ ♠å✐ x ∈ X t❤➻ ❢ ✈➔ ❣ ❝â ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤✉♥❣ ❞✉② ♥❤➜t✳ ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sỷ p : X ì X R ợ p(x, y) = d(x, y) ợ (x, y) X ì X ✳ ❑❤✐ ✤â✱ t❤❡♦ ❱➼ ❞ö ✷✳✶✳✷✱ p ❧➔ ω ✲❦❤♦↔♥❣ ❝→❝❤ tr➯♥ X ✳ ❚❤❡♦ ❣✐↔ t❤✐➳t t❤➻ ỵ ✤÷đ❝ t❤ä❛ ♠➣♥✳ ●✐↔ sû {xn} ❧➔ ❞➣② tr♦♥❣ X s❛♦ ❝❤♦ lim p(t, f xn ) = lim p(t, gxn ) = 0, n→∞ n→∞ ✭✷✮ tù❝ lim d(t, f xn ) = lim d(t, gxn ) = n→∞ ❉♦ ✤â n→∞ lim f xn = lim gxn = t n→∞ n→∞ ❱➻ f ✈➔ g ❧➔ ❤❛✐ →♥❤ ①↕ t÷ì♥❣ t❤➼❝❤ ♥➯♥ ✭✸✮ lim d(f gxn , gf xn ) = n→∞ ❚ø ✭✷✮ ✈➔ ✭✸✮ s r ợ > tỗ t số tü ♥❤✐➯♥ nε s❛♦ ❝❤♦ max{d(t, f xn ), d(t, gxn ), d(f gxn , gf xn )} < ε ✈ỵ✐ ♠å✐ n > nε ❉♦ ✤â lim max{d(t, f xn ), d(t, gxn ), d(f gxn , gf xn )} = n→∞ ◆❤÷ ✈➟②✱ ✤✐➲✉ ❦✐➺♥ ✭✐✐✐✮ tr♦♥❣ ỵ ữủ tọ t ự tr ỵ ụ ữủ t❤ä❛ ♠➣♥✳ ●✐↔ sû ✤✐➲✉ ❦✐➺♥ ✭✐✈✮ ❦❤ỉ♥❣ ✤÷đ❝ t❤ä❛ õ tỗ t u X s u = f u ❤♦➦❝ u = gu ♥❤÷♥❣ ✐♥❢{d(u, f x) + d(u, gx) + d(f gx, gf x) : x ∈ X} = ❚ø ✤â s✉② r❛ ợ ộ n = 1, 2, tỗ t↕✐ xn ∈ X s❛♦ ❝❤♦ d(u, f xn ) < 1 , d(u, gxn ) < , d(f gxn , gf xn ) < n n n õ tỗ t {xn} tr X s ❝❤♦ lim d(u, f xn ) = lim d(u, gxn ) = lim d(f gxn , gf xn ) = n→∞ n→∞ n→∞ ✭✹✮ ✣✐➲✉ ♥➔② ❝❤ù♥❣ tä f xn → u, gxn → u✳ ❱➻ f ✈➔ g ❧✐➯♥ tö❝ ♥➯♥ gf xn → gu, f gxn → f u ❉♦ ✤â lim d(f gxn , gf xn ) = d(f u, gu) n→∞ ❑➳t ❤đ♣ ✈ỵ✐ ✭✹✮ t❛ ❝â d(f u, gu) = 0✱ tù❝ ❧➔ f u = gu✳ ❉♦ ✤â gu = u✳ ❚❤❡♦ ✭✶✮ t❛ ❝â d(t, gxn ) ≤ rd(t, f xn ) + ϕ(f xn ) − ϕ(gxn ) ✈ỵ✐ ♠å✐ n ❑➳t ❤đ♣ ✈ỵ✐ t➼♥❤ ❧✐➯♥ tư❝ ❝õ❛ d ✈➔ ϕ s✉② r❛ o ≤ d(t, u) ≤ rd(t, u) + ϕ(u) − ϕ(u) = rd(t, u) ❱➻ r ∈ [0, 1) ♥➯♥ tø ❜➜t ✤➥♥❣ t❤ù❝ tr➯♥ s✉② r❛ d(t, u) = 0✱ tù❝ ❧➔ t = u✳ ❚❤❛② t = u ✈➔ x = u ✈➔♦ ✭✶✮ ✈➔ sû ❞ö♥❣ f u = gu t❛ ❝â d(u, gu) = 0✳ ✣✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥ ✈ỵ✐ gu = u✳ ❉♦ õ ỵ ữủ tọ t ỵ f g õ t t p ỵ ❈❤♦ (X, d) ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ✤➛② ✤õ ✈ỵ✐ ω✲❦❤♦↔♥❣ ❝→❝❤ ❤❛✐ →♥❤ ①↕ f, g : X−→ X ✈➔ ❤❛✐ ❤➔♠ sè ϕ, ψ tø X−→ [0, ∞) s tr ỵ ✷✳✶✳✹ ✤÷đ❝ t❤ä❛ ♠➣♥ ✈➔ ❣✐↔ sû r➡♥❣ ✭✐✮ ◆➳✉ {xn}n∈N ❧➔ ❞➣② tr♦♥❣ X ♠➔ lim f xn = lim gxn = t n→∞ n→∞ ✈ỵ✐ t t❤✉ë❝ ❳ t❤➻ lim max{p(t, f xn ), p(t, gxn ), p(f gxn , gf xn )} = n→∞ ✷✺ ✭✐✐✮ p(gx, gy) ≤ a1 p(f x, f y) + a2 p(f x, gx) + a3 p(f y, gy) + a4 p(f x, gy) + a5 [p(gx, f y)d(f y, gx)] + [ϕ(f x) − ϕ(gx)] + [ψ(f y) − ψ(gy)] ✈ỵ✐ ♠å✐ x, y ∈ X, a1 , a2 , a3 , a4 ✈➔ a5 ∈ [0, 1) ✈ỵ✐ a1 + a4 + a5 < ✈➔ a1 + a2 + a3 + 2a4 < ❑❤✐ ✤â✱ ❢ ✈➔ ❣ ❝â ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤✉♥❣ ❞✉② ♥❤➜t✳ ự sỷ x0 tũ ỵ X ✳ ❚❤❡♦ ✷✳✶✳✹✭✐✮✱ t❛ ①➙② ❞ü♥❣ ✤÷đ❝ ❞➣② {xn} tr♦♥❣ X s❛♦ ❝❤♦ gxn−1 = f xn ✈ỵ✐ n = 1, 2, ❈❤♦ γn = p(f xn , f xn+1 ) ✈ỵ✐ n ≥ 0✳ ❚ø ✭✐✐✮ t❛ ❝â γj+1 = p(gxj , gxj+1 ) ≤ a1 p(f xj , f xj+1 ) + a2 p(f xj , gxj ) + a3 p(f xj+1 , gxj+1 ) + a4 p(f xj , gxj+1 ) + a5 [p(gxj , f xj+1 )d(f xj+1 , gxj )] + [ϕ(f xj ) − ϕ(gxj )] + [ψ(f xj+1 ) − ψ(gxj+1 )] ≤ (a1 + a2 + a4 )γj + (a3 + a4 )γj+1 + [ϕ(f xj ) − ϕ(f xj+1 )] + [ψ(f xj+1 ) − ψ(f xj+2 )] ❉♦ ✤â γj+1 ≤ L1 γj + L2 [ϕ(f xj ) − ϕ(f xj+1 ) + ψ(f xj+1 ) − ψ(f xj+2 )] ❚r♦♥❣ ✤â ❱➻ ✈➟② L1 = a1 + a2 + a4 , − a3 − a4 γj ≤ L1 L2 γ0 + [ϕ(f x0 ) + ψ(f x1 )] − L1 − L1 n j=1 ✈ỵ✐ ♠å✐ n ≥ 1✳ ❉♦ ✤â ∞ γn n=1 L2 = − a3 − a4 ❤ë✐ tư✳ ❱ỵ✐ n, r ≥ 1✱ t❛ ❝â n+r−1 p(f xn , f xn+r ) ≤ γi i=n ✷✻ ❚❤❡♦ ❇ê ✤➲ ✷✳✶✳✸✱ {f xn}∞ n=1 ❧➔ ❞➣② ❈❛✉❝❤② tr♦♥❣ X ✳ ❱➻ X ❧➔ ởt ổ tr tỗ t t ∈ X s❛♦ ❝❤♦ f xn−→ t ❦❤✐ n → ∞✳ ❚ø ✤✐➲✉ ❦✐➺♥ ✭✐✮✱ t❛ ❝â lim max{p(t, f xn ), p(t, gxn ), p(f gxn , gf xn )} = n→∞ ●✐↔ sû r➡♥❣ t = f t ❤♦➦❝ t = gt✳ ❑❤✐ ✤â✱ ✤✐➲✉ ❦✐➺♥ ✭✐✈✮ ỵ t õ < {p(t, f x) + p(t, gx) + p(f gx, gf x) : x ∈ X} ≤ ✐♥❢{p(t, f xn ) + p(t, gxn ) + p(f gxn , gf xn ) : n ∈ N} = ✣➙② ❧➔ ♠ët ✤✐➲✉ ♠➙✉ t❤✉➝♥✳ ❉♦ ✤â t ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤✉♥❣ ❝õ❛ f ✈➔ g✳ ❚ø ❇ê ✤➲ ✷✳✶✳✸ ✈➔ ✤✐➲✉ ❦✐➺♥ ✭✐✐✮ s✉② r❛ t ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤✉♥❣ ❞✉② ♥❤➜t ❝õ❛ f ✈➔ g ✷✳✷✳ ❙Ü ❚➬◆ ❚❸■ ✣■➎▼ ❇❻❚ ✣❐◆● ❈Õ❆ ❈⑩❈ ⑩◆❍ ❳❸ ✣❆ ❚❘➚ ▼ö❝ ♥➔② tr ởt số ỵ sỹ tỗ t ✤✐➸♠ ❜➜t ✤ë♥❣ ✈➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤✉♥❣ ❝õ❛ ❝→❝ →♥❤ ①↕ ✤❛ trà tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ✤➛② ✤õ ✈ỵ✐ ω ✲❦❤♦↔♥❣ ❝→❝❤✳ ❚r♦♥❣ ♠ư❝ ♥➔② t❛ ❣✐↔ sû (X, d) ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ✤➛② ✤õ ✈ỵ✐ ω✲❦❤♦↔♥❣ ❝→❝❤ p✱ M ❧➔ t➟♣ ❝♦♥ ✤â♥❣ ❝õ❛ X ✈➔ Cl(M ) ❧➔ ❤å ❝→❝ t➟♣ ❝♦♥ ✤â♥❣ ❦❤→❝ ré♥❣ ❝õ❛ M ✳ ✷✳✷✳✶✳ ✣à♥❤ ♥❣❤➽❛✳ ✶✮ ⑩♥❤ ①↕ ✤ì♥ trà f : X−→ X ✤÷đ❝ ❣å✐ ❧➔ ω✲❦❛♥❛♥ ♥➳✉ tỗ t ởt tr X r 0, s❛♦ ❝❤♦ ✈ỵ✐ ♠é✐ x, y ∈ X, ω(f (x), f (y)) ≤ r{ω(x, f (x)) + ω(y, f (y))} ✷✮⑩♥❤ ①↕ ✤❛ trà T : X−→ 2X ữủ K tỗ t ♠ët sè ❦❤æ♥❣ ➙♠ r ∈ 0, 21 ✈➔ ♠ët ω✲❦❤♦↔♥❣ ❝→❝❤ s❛♦ ❝❤♦ ✈ỵ✐ ❜➜t ❦ý x ∈ X, u T (x) y X tỗ t v ∈ T (y) s❛♦ ❝❤♦ ω(x, v) ≤ r{ω(x, u) + ω(y, v)} ✷✼ ✣➦❝ ❜✐➺t✱ ♥➳✉ ❧➜② ω = d t❤➻ →♥❤ ①↕ ω✲❦❛♥❛♥ ❧➔ co✲❦❛♥❛♥✳ ✷✳✷✳✷✳ ✣à♥❤ ỵ T : M Cl(M ) ởt Kω ✲→♥❤ ①↕ ✤❛ trà s❛♦ ❝❤♦ ✐♥❢{p(x, y) + p(x, T x) : x ∈ X} > ✈ỵ✐ ♠å✐ u ∈ X ♠➔ u ∈/ T (u) t❤➻ T ❝â ✤✐➸♠ ❜➜t ✤ë♥❣✳ ❈❤ù♥❣ ♠✐♥❤✳ ❈❤♦ u0 ❧➔ ởt tỷ tũ ỵ M u1 ∈ T (u0) ❧➔ ❝è ✤à♥❤✳ ❚ø T ❧➔ ♠ët K tỗ t u2 T (u1) s ❝❤♦ p(u1 , u2 ) ≤ rp(u0 , u1 ) + rp(u1 , u2 ), tr♦♥❣ ✤â r ∈ 0, ✈➔ ❞♦ ✤â p(u1 , u2 ) ≤ r p(u0 , u1 ) 1r tử ỵ t÷ì♥❣ tü✱ t❛ ①➙② ❞ü♥❣ ✤÷đ❝ ❞➣② {un} tr♦♥❣ M s❛♦ ❝❤♦ ✈ỵ✐ ♠å✐ n ∈ N, un+1 ∈ T (un) ✈➔ p(un , un+1 ) ≤ r p(un−1 , un ); 1−r ✈ỵ✐ r ❝è ✤à♥❤✱ < r < 21 ✳ ❱ỵ✐ ❜➜t ❦ý n ∈ N✱ t❛ ❝â p(un , un+1 ) ≤ ✣➦t r 1−r n p(u0 , u1 ) t❤➻ < λ < ợ m n ữỡ m > n t❛ ❝â λ= r 1−r p(un , um ) ≤ p(um , un+1 ) + p(un+1 , un+2 ) + · · · + p(um−1 , um ); ≤ λn p(u0 , u1 ) + λn+1 p(u0 , u1 ) + · · · + λm−1 p(u0 , u1 ); rn ≤ p(u0 , u1 ) 1−r ❚ø ✤â s✉② r❛ p(un, um) → ❦❤✐ n → ∞✳ ❚❤❡♦ ❇ê ✤➲ ✷✳✶✳✸✱ {un} ❧➔ ♠ët ❞➣② ❈❛✉❝❤②✳ ❚ø X ❧➔ ✤➛② ✤õ t❛ ♥❤➟♥ ✤÷đ❝ {un} ❤ë✐ tư ✈➲ v0 ∈ X ✳ M ❧➔ t➟♣ ✤â♥❣ ✷✽ ♥➯♥ v0 ∈ M ✳ ❈❤♦ ❝è ✤à♥❤ n ∈ N ✳ ❚ø {un} ❤ë✐ tư tỵ✐ v0 ✈➔ p(un, ) ỷ tử ữợ t õ p(un , v0 ) ≤ lim inf p(un , um ) ≤ m→∞ rn p(u0 , u1 ) 1−r ❱➻ ✈➟②✱ ❦❤✐ n → ∞ t❛ ❝â (un, v0) → 0✳ ●✐↔ sû r➡♥❣ v0 ∈ T (v0)✳ ❑❤✐ ✤â✱ tø ❣✐↔ t❤✐➳t t❛ ❝â < ✐♥❢{p(un , v0 ) + p(un , T (un )) : n ∈ N} ≤ ✐♥❢{p(un , v0 ) + p(un , T (un )) : n ∈ N} ≤ ✐♥❢{p(un , v0 ) + p(un , T (un+1 )) : n ∈ N} rn ≤ ✐♥❢ p(u0 , u1 ) + λn p(u0 , u1 ) : n ∈ N = 1−r ✣✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥ ✈➔ ❞♦ ✤â v0 ∈ T (v0)✱ tù❝ v0 ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ T ✷✳✷✳✸✳ ✣à♥❤ ♥❣❤➽❛✳ ●✐↔ sû T : M −→ Cl(M )✳ ❉➣② {xn} tr M ữủ ố ợ T ✱ ♥â✐ ❣å♥ ❧➔ ❞➣② ❧➦♣ ♥➳✉ un ∈ T (un1) ợ n ỵ ộ Kω ✲→♥❤ ①↕ T : M −→ Cl(M ) ❝â ✤✐➸♠ ❜➜t ✤ë♥❣ ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ ❧➔ ✈ỵ✐ ❜➜t ❦ý ❞➣② ❧➦♣ {un} tr♦♥❣ M, {p(u0 , un )} ❤ë✐ tö ✈➲ 0✳ un → v0 ∈ M ✱ ❞➣② số tỹ ự ự ỵ tỗ t ởt {un} s un v0 ∈ M ✈ỵ✐ p(un , v0 ) ≤ lim inf p(un , um ) ≤ m→∞ ✈➔ rn p(u0 , u1 ) 1−r p(un , un+1 ) ≤ λn p(u0 , u1 ), tr♦♥❣ ✤â λ = −r r < ú ỵ r p(un, v0) ❦❤✐ n → ∞ ❚❤➯♠ ♥ú❛✱ tø un ∈ T (un−1) ✈➔ T ❧➔ ♠ët Kω ✲→♥❤ ①↕✱ ❝â ∈ T (v0) s❛♦ ❝❤♦ p(un , ) ≤ r(p(un−1 , un ) + p(v0 , )) ≤ rp(un−1 , un ) + rp(v0 , un ) + rp(un , ) r r ≤ p(un−1 , un ) + p(v0 , un ) 1−r 1−r ✷✾ ✈➔ ❞♦ ✤â p(un, vn) → ❦❤✐ n → ∞✳ ❉♦ ✤â tø ❇ê ✤➲ ✷✳✶✳✸✱ t❛ ❝â → v0 ✈➔ tø ∈ T (v0) ❧➔ t➟♣ ✤â♥❣ s✉② r❛ v0 ∈ T (v0) ❇➙② ❣✐í t❛ ❝❤ù♥❣ ởt số t q sỹ tỗ t t ỵ {Tn} ♠ët ❞➣② ❝→❝ →♥❤ ①↕ ✤❛ trà tø ▼ ✈➔♦ Cl(M ) sỷ r tỗ t số ≤ r < 21 s❛♦ ❝❤♦ ✈ỵ✐ ❜➜t ❦ý ❤❛✐ →♥❤ ①↕ Ti, Tj ∈ {Tn } ✈➔ ✈ỵ✐ ❜➜t ý x M, u Ti (x) tỗ t v ∈ Tj (y) ✈ỵ✐ ♠å✐ y ∈ M ♠➔ p(u, v) ≤ r(p(x, u) + p(y, v)), ✈➔ ✈ỵ✐ ♠é✐ n ≥ ✐♥❢{p(x, u) + p(x, Tn(x)) : x ∈ X} > 0, ✈ỵ✐ ❜➜t ❦ý u ∈/ Tn(u)✳ ❑❤✐ ✤â✱ {Tn} ❝â ♠ët ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤✉♥❣✳ ự u0 ởt tỷ tũ ỵ M u1 T1(u0) õ tỗ t u2 ∈ T2(u1) s❛♦ ❝❤♦ p(u1 , u2 ) ≤ r p(u0 , u1 ) 1r tỗ t ♠ët ❞➣② {un} s❛♦ ❝❤♦ un+1 ∈ Tn+1(un) ✈➔ ✈ỵ✐ ♠å✐ n ≥ 1✱ p(un , un+1 ) ≤ r 1−r n p(u0 , u1 ) ✣➦t λ = r r ú ỵ r < < ✈➔ p(un , un+1 ) ≤ λn p(u0 , u1 ), ✈ỵ✐ ♠å✐ n ≥ 1✳ ❈❤♦ n → ∞✱ t❛ ♥❤➟♥ ✤÷đ❝ ❞➣② {un} ❧➔ ♠ët ❞➣② ❈❛✉❝❤② tr♦♥❣ X ✳ ✣➦t p = lim un tr♦♥❣ M ✳ ❇➙② ❣✐í t❛ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ p ∈ Tn (p)✳ ❈❤♦ n→∞ n≥1 Tm ❧➔ ♠ët ♣❤➛♥ tû tò② þ ❝õ❛ {Tn }✳ ❚ø un ∈ Tn (un−1 ) tt s r tỗ t sn Tm(p) s❛♦ ❝❤♦ p(un , sn ) ≤ r(p(un−1 , un ) + p(p, sn )) ✸✵ ❚❛ ❝❤ù♥❣ ♠✐♥❤ t÷ì♥❣ tỹ ữ tr ỵ ữủ rn p(un , p) ≤ lim inf p(un , um ) ≤ p(u0 , u1 ) m→∞ 1−r ✈➔ ✈➳ ♣❤↔✐ ❤ë✐ tư ✈➲ ❦❤✐ n → ∞✳ ❇➙② ❣✐í ❣✐↔ sû r➡♥❣ p ∈/ Tm(p)✳ ❑❤✐ ✤â✱ t❤❡♦ ❣✐↔ t❤✐➳t ✈➔ ✈ỵ✐ n > m ✈➔ m ≥ t❛ ❝â ≤ ✐♥❢{p(u, p) + p(u, Tm (u)) : u ∈ X} ≤ ✐♥❢{p(um−1 , p) + p(um−1 , Tm (um−1 )) : m ∈ N} ≤ ✐♥❢{p(um−1 , p) + p(um−1 , um ) : m ∈ N} rm−1 ≤ ✐♥❢ p(u0 , u1 ) + λm−1 p(u0 , u1 ) : m ∈ N = 1−r ✣✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥ ✈➔ ❞♦ ✤â p ∈ Tm(p) ữ Tm ởt tỷ tũ ỵ {Tn} ♥➯♥ p ❧➔ ♠ët ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤✉♥❣ ❝õ❛ Tn ự tữỡ tỹ ữ ỵ t õ ỵ s ỵ {Tn} ♠ët ❞➣② ❝→❝ →♥❤ ①↕ ✤❛ trà tø M ✈➔♦ Cl(M ) sỷ r tỗ t ởt số r ✈ỵ✐ ≤ r < 21 ✈➔ s❛♦ ❝❤♦ ✈ỵ✐ ❜➜t ❦ý ❤❛✐ →♥❤ ①↕ Ti, Tj ✈➔ ✈ỵ✐ t ý x M, u Ti(x) tỗ t v ∈ Tj (y) ✈ỵ✐ t➜t ❝↔ y ∈ M ♠➔ p(u, v) ≤ r(p(x, u) + p(y, v)) t❤➻ {Tn} ❧➔ ♠ët ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤✉♥❣ ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ ❜➜t ❦ý ❞➣② ❧➦♣ {un} tr♦♥❣ ▼✱ un → v0 ∈ M ✱ ❞➣② sè t❤ü❝ {p(v0, un)} ❤ë✐ tö ✈➲ 0✳ ✸✶ ❑➌❚ ▲❯❾◆ ▲✉➟♥ ✈➠♥ ✤➣ ✤↕t ✤÷đ❝ ❝→❝ ❦➳t q✉↔ ❝❤➼♥❤ s❛✉ ✤➙②✿ ✶✳ ❚r➻♥❤ ❜➔② ♠ët õ tố ởt số ỵ sỹ tỗ t t t ❝õ❛ ❝→❝ →♥❤ ①↕ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ o✲♠➯tr✐❝✳ ✷✳ ❚r➻♥❤ ❜➔② ❝→❝ ❦❤→✐ ♥✐➺♠ ω✲❦❤♦↔♥❣ ❝→❝❤ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ởt số ỵ sỹ tỗ t ❜➜t ✤ë♥❣ ❝õ❛ ❝→❝ →♥❤ ①↕ ✤ì♥ trà ✈➔ ✤❛ trà tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ✤➛② ✤õ ✈ỵ✐ ω✲❦❤♦↔♥❣ ❝→❝❤✳ ✸✳ ❈❤ù♥❣ ♠✐♥❤ ❝❤✐ t✐➳t ♠ët sè ❦➳t q✉↔ ♠➔ tr♦♥❣ ❝→❝ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❝❤ù♥❣ ♠✐♥❤ ✈➢♥ t➢t ❤♦➦❝ ❜ä q✉❛ ❝❤ù♥❣ ♠✐♥❤✳ ✹✳ ✣÷❛ r❛ ❱➼ ❞ư ✷✳✶✳✷✱ ♠✐♥❤ ❤å❛ ❝❤♦ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ✈ỵ✐ ω✲❦❤♦↔♥❣ ❝→❝❤✳ ữ r ởt t q ợ sỹ tỗ t ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤✉♥❣ ❝õ❛ ❤❛✐ →♥❤ ①↕ ❧✐➯♥ tö❝ tữỡ t tr ổ tr ợ õ ỵ ❬✶❪ ✣✐♥❤ ❍✉② ❍♦➔♥❣✱ ▲➯ ❑❤→♥❤ ❍÷♥❣✱ ❇ị✐ ❚❤à ❚❤ó② ❱✐♥❤ ✭✷✵✵✾✮✱ ✣✐➸♠ ❜➜t ✤ë♥❣ ❝❤✉♥❣ ❝õ❛ ❝→❝ →♥❤ ①↕ t÷ì♥❣ t❤➼❝❤ ②➳✉ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ o✲♠➯tr✐❝✱ ❚↕♣ ❝❤➼ ❑❤♦❛ ❤å❝ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❱✐♥❤✱ ❚➟♣ ❳❳❳❱■■■✱ ❙è ✶❆✱ ✸✶ ✲ ✸✽✳ ❬✷❪ ✣✐♥❤ ❍✉② ❍♦➔♥❣✱ P❤❛♥ ❆♥❤ ❚➔✐✱ ◆❣✉②➵♥ ✣➻♥❤ ▲➟♣ ✭✷✵✵✾✮✱ ▼ët sè t➼♥❤ ❝❤➜t ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ♦✲♠➯tr✐❝ ✈➔ ♦✲♠➯tr✐❝ ♠↕♥❤✱ ❚↕♣ ❝❤➼ ❑❤♦❛ ❤å❝ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❱✐♥❤✱ ❚➟♣ ❳❳❳❱■■■✱ ❙è ✷❆✱ ✷✻ ✲ ✸✸✳ ❬✸❪ ▼✳ ❆❛♠r✐ ❛♥❞ ❉✳ ❊■ ▼♦✉t❛✇❛❦✐❧ ✭✷✵✵✸✮✱ ❈♦♠♠♦♥ ❢✐①❡❞ ♣♦✐♥t ✉♥❞❡r ❝♦♥✲ tr❛❝t✐✈❡ ❝♦♥❞✐t✐♦♥s ✐s s②♠♠❡tr✐❝ s♣❛❝❡s✱ ❆♣♣❧✐❡❞ ▼❛t❤❡♠❛t✐❝s ❊ ✲ ◆♦t❡s✱ ◆♦✳✸✱ ♣♣✳✶✺✻ ✲ ✶✻✷✳ ❬✹❪ ❖✳ ❑❛❞❛✱ ❚✳ ❙✉③✉❦✐ ❛♥❞ ❲✳ ❚❛❦❛❤❛s❤✐ ✭✶✾✾✻✮✱ ◆♦♥❝♦♥✈❡① ♠✐♥✐♠✐③❛t t❤❡✲ ♦r❡♠s ❛♥❞ ❢✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠s ✐♥ ❝♦♠♣❧❡t❡ ♠❡tr✐❝ s♣❛❝❡s✱ ▼❛❥❛♣♦♥✱ ✹✹✱ ✸✽✶ ✲ ✸✾✶✳ ❬✺❪ ❆❜❞✉❧ ▲❛t✐❢ ❛♥❞ ❲❛❢❛❛ ❆✳ ❆❧❜❛r ✭✷✵✵✼✮✱ ❋✐①❡❞ ♣♦✐♥t r❡s✉❧ts ▼✉❧t✐✈❛❧✉❡❞ ▼❛♣s✱ ■♥t✳ ❏✳ ❈♦♥t❡♠♣✳ ▼❛t❤ ❙❝✐❡♥❝❡s✱ ❱♦❧✳✷✱ ♥♦✳ ✶✶✷✾ ✲ ✶✶✸✻✳ ❬✻❪ ❏❡♦♥❣ ❙❤❡♦❦ ❯♠❡ ✭✷✵✶✵✮✱ ❊①✐st❡♥❝❡ t❤❡♦r❡♠s ❢♦r ●❡♥❡r❛❧✐③❡❞ ❞✐st❛♥❝❡ ♦♥ ❝♦♠♣❧❡t❡ ♠❡tr✐❝ s♣❛❝❡r✱ ❍✐♥❞❛♠ ♣✉❜❧✐s❤✐♥❣ ❝♦r♣♦r❛t✐♦♥✱ ❋✐①❡❞ ♣♦✐♥t t❤❡✲ ♦r② ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s✱ ❆rt✐❝❧❡ ■❉ ✸✾✼✶✺✵✱ ✷✶ ♣❛❣❡s✱ ❞♦✐✿ ✶✵✳✶✶✺✺✴✸✾✼✶✺✵✳ ✸✸ ... ❣å✐ ❧➔ ♠ët o? ??♠➯tr✐❝ ♠↕♥❤ ♥➳✉ d ❧➔ o? ??♠➯tr✐❝ ✈➔ ✈ỵ✐ ♠é✐ x ∈ X ✱ ✈ỵ✐ ♠é✐ r > ❤➻♥❤ ❝➛✉ X−→ R B(x, r) = {y ∈ X : d(x, y) < r} ❧➔ ♠ët ❧➙♥ ❝➟♥ ❝õ❛ x✳ ❑❤æ♥❣ ❣✐❛♥ tổổ X ũ ợ ởt otr d tr õ ữủ ổ otr ỵ ... o? ??♠➯tr✐❝ ✈ỵ✐ d(X) < ∞✳ ✶✮ X ✤÷đ❝ ❣å✐ ❧➔ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ (W 3) ♥➳✉ ✈ỵ✐ x, y ✈➔ ❞➣② {xn} tr♦♥❣ X ♠➔ lim d(x, xn ) = lim d(y, xn ) = t❤➻ x = y ❀ n→∞ n→∞ ✷✮ X ✤÷đ❝ ❣å✐ ❧➔ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ (W. .. ự ỵ tt t ự sỹ tỗ t t tr ổ otr sỹ tỗ t t tr ổ ❣✐❛♥ ♠➯tr✐❝ ✈ỵ✐ ω ✲❦❤♦↔♥❣ ❝→❝❤✳ ❱ỵ✐ ♠ư❝ ✤➼❝❤ ✤â✱ ❧✉➟♥ ✈➠♥ ✤÷đ❝ ❝❤✐❛ ❧➔♠ ❤❛✐ ❝❤÷ì♥❣ ❈❤÷ì♥❣ ✶✳ ❙ü tỗ t t tr ổ otr r ❝❤÷ì♥❣ ♥➔②✱ ❝❤ó♥❣ tỉ✐ tr➻♥❤