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TR×ÍNG ��I HÅC VINH Vi»n S÷ Ph¤m Tü Nhi¶n �������o0o������� B�I T�P TI�U LU�N T�P MÐ, PH�N TRONG V� C�C V�N �� LI�N QUAN Chuy¶n ng nh Lþ thuy¸t XS v TK To¡n Gi£ng vi¶n NGUY�N THÀ QUÝNH TRANG Håc vi¶n NGUY�N THÀ NGÅC ANH Lîp K28 To¡n Vinh, 32021 1 GIÎI THI�U, ��T V�N �� 1 Giîi thi»u, �°t v§n � 1 1 Giîi thi»u Tªp hñp mð, hay tªp mð, ph¦n trong l kh¡i ni»m cì b£n trong topo Nâ công �÷ñc sû döng trong c¡c l¾nh vüc kh¡c cõa to¡n håc, trong c¡c khæng gian kh¡c câ thº topo hâa C¡c kh¡i ni»m n y l têng.
ì ữ P ỹ ❇⑨■ ❚❾P ❚■➎❯ ▲❯❾◆ ❚❾P ▼Ð✱ P❍❺◆ ❚❘❖◆● ❱⑨ ❈⑩❈ ỵ tt ❚♦→♥ ●✐↔♥❣ ✈✐➯♥✿ ❍å❝ ✈✐➯♥ ✿ ▲ỵ♣✿ ◆●❯❨➍◆ ❚❍➚ ◗❯Ý◆❍ ❚❘❆◆● ◆●❯❨➍◆ ❚❍➚ ◆●➴❈ ❆◆❍ ❑✷✽ ❚♦→♥ ❱✐♥❤✱ ✸✴✷✵✷✶ ✶✳ ●■❰■ ❚❍■➏❯✱ ✣➄❚ ❱❻◆ ✣➋ ✶ ●✐ỵ✐ t❤✐➺✉✱ ✤➦t ✈➜♥ ✤➲ ✶✳✶ ●✐ỵ✐ t❤✐➺✉ ❚➟♣ ❤đ♣ ♠ð✱ ❤❛② t➟♣ ♠ð✱ ♣❤➛♥ tr♦♥❣ ❧➔ ❦❤→✐ ♥✐➺♠ ❝ì ❜↔♥ tr♦♥❣ t♦♣♦✳ ◆â ❝ơ♥❣ ✤÷đ❝ sû ❞ư♥❣ tr♦♥❣ ❝→❝ ❧➽♥❤ ✈ü❝ ❦❤→❝ ❝õ❛ t♦→♥ ❤å❝✱ tr♦♥❣ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ ❦❤→❝ ❝â t❤➸ t♦♣♦ ❤â❛✳ ❈→❝ ❦❤→✐ ♥✐➺♠ ♥➔② ❧➔ tê♥❣ q✉→t tø ❦❤→✐ ♥✐➺♠ ♠✐➲♥ tr♦♥❣ ❝õ❛ ♠ët t➟♣ ❤ñ♣ ✤✐➸♠ tr♦♥❣ ❤➻♥❤ ❤å❝ ✈➔ tr♦♥❣ ❣✐↔✐ t➼❝❤✳ ❱➼ ❞ö ✶✳ ❈→❝ ✤✐➸♠ (x, y) t❤ä❛ ♠➣♥ x2 + y2 = r2 ❧➔ ✤÷í♥❣ trá♥ t➙♠ ❖✭✵✱✵✮ ❜→♥ ❦➼♥❤ r✳ ❈→❝ ✤✐➸♠ (x, y) t❤ä❛ ♠➣♥ x2 + y < r2 ❧➔ ❝→❝ ✤✐➸♠ ❜➯♥ tr♦♥❣ ❤➻♥❤ trá♥✳ ❚➟♣ ❝→❝ ✤✐➸♠ ❜➯♥ tr♦♥❣ ❤➻♥❤ trá♥ ❧➔ t➟♣ ♠ð✱ t➟♣ ❝→❝ ✤✐➸♠ tr➯♥ ✤÷í♥❣ trá♥ ❧➔ t➟♣ ✤â♥❣✳ ❍đ♣ ❝õ❛ ❝→❝ ✤✐➸♠ ❜➯♥ tr♦♥❣ ✈➔ tr➯♥ ✤÷í♥❣ trá♥ ❧➔ t➟♣ ✤â♥❣✳ ❚➟♣ ❆ ❧➔ t➟♣ ♠ð tr♦♥❣ X ♥➳✉ ❆ ⊆ X ✈➔ X ❝ô♥❣ ♠ð tr♦♥❣ ❆ ✈➔ ❤✐➸♥ ♥❤✐➯♥ ❆ ♠ð tr♦♥❣ ❝❤➼♥❤ ♥â ✈➻ ❆ ❂ ❆✳ ✶✳✷ ✣➦t ✈➜♥ ✤➲ ✶✳✷✳✶✳ ❱❛✐ trá ❝ỉ♥❣ ❝ư ❝õ❛ ❝→❝ ❦❤→✐ ♥✐➺♠ tr♦♥❣ ❣✐↔✐ t➼❝❤ ❚➟♣ ♠ð✱ ♣❤➛♥ tr♦♥❣✱✳✳ ❧➔ ❝→❝ ❦❤→✐ ♥✐➺♠ ❝ì ❜↔♥ ✈➔ ①✉➜t ❤✐➺♥ ❤➛✉ ❤➳t tr♦♥❣ ❝→❝ ❧➽♥❤ ✈ü❝ ❝õ❛ ❣✐↔✐ t➼❝❤ ♥❤÷ ❚♦♣♦ ✤↕✐ ❝÷ì♥❣✱ ❣✐↔✐ t t ỗ t ự ◗✉② ❤♦↕❝❤ ♣❤✐ t✉②➳♥✱ ●✐↔✐ t➼❝❤ ♣❤ù❝✱✳✳ ❱➻ ✈➟② ✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉ tr✐ t❤ù❝ ❧✉➟♥ ✈➲ ❝→❝ ❦❤→✐ ♥✐➺♠ ♥➔② t❤ü❝ sü ❝➛♥ t❤✐➳t tr♦♥❣ ✈✐➺❝ ❞↕② ❤å❝ ❝→❝ ♠æ♥ t ỗ t ỳ q✉❛♥ ♥✐➺♠ s❛✐ ❧➛♠ ❝õ❛ s✐♥❤ ✈✐➯♥ ✈➲ ❦❤→✐ ♥✐➺♠ t➟♣ ♠ð ❚r♦♥❣ ❤❛✐ t❤→♥❣ ✾ ✈➔ ✶✵✴✷✵✶✼✱ ♠ët t❤ü❝ st ữợ ọ trỹ t ữủ t✐➳♥ ❤➔♥❤ tr➯♥ ✶✵ s✐♥❤ ✈✐➯♥ ♥➠♠ t❤ù ✸ ♥❣❤➔♥❤ s÷ ♣❤↕♠ t♦→♥ ð ❝→❝ tr÷í♥❣ ✤↕✐ ❤å❝✿ ❙➔✐ ❣á♥✱ tỹ ố ỗ ❦❤→✐ ♥✐➺♠ t➟♣ ♠ð✳ ❈→❝ s✐♥❤ ✈✐➯♥ ♥➔② ✤➣ ❦➳t t❤ó❝ ❝→❝ ❤å❝ ♣❤➛♥ ❦❤ỉ♥❣ ❣✐❛♥ tỉ♣♦ ✈➔ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ð ♥➠♠ t❤ù ❤❛✐ ✤↕✐ ❤å❝✳ ▼ö❝ ✤➼❝❤ ❝õ❛ ✈✐➺❝ ❦❤↔♦ s→t ❧➔ ♥❤➡♠ t➻♠ ❤✐➸✉ q✉❛♥ ♥✐➺♠ ❝õ❛ s✐♥❤ ✈✐➯♥ ✈➲ t➟♣ ♠ð s❛✉ ❦❤✐ ✤➣ ❤å❝ ①♦♥❣ ❝→❝ ❤å❝ ♣❤➛♥ tr➯♥✳ ❚➟♣ ♠ð tr♦♥❣ ◆❣✉②➵♥ ❚❤à ◆❣å❝ ❆♥❤ ✶ ❈❛♦ ❤å❝ ❑❤â❛ ✷✽ ❚♦→♥ ✶✳ ●■❰■ ❚❍■➏❯✱ ✣➄❚ ❱❻◆ ✣➋ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ t❤❡♦ ❜❛ ❝→❝❤✿ ✣à♥❤ ♥❣❤➽❛ t❤❡♦ ❤➻♥❤ ❝➛✉ ♠ð✱ ✤à♥❤ ♥❣❤➽❛ t❤❡♦ ♣❤➛♥ tr♦♥❣ ✈➔ ✤à♥❤ ♥❣❤➽❛ t❤❡♦ ❧➙♥ ❝➟♥✭ ❚r➻♥❤ ❜➔② ✤à♥❤ ♥❣❤➽❛ s➩ ✤÷đ❝ ❝ư t❤➸ ð ♣❤➛♥ s❛✉✮✳ ❑➳t q✉↔ ❝❤♦ t❤➜② ❝→❝ s✐♥❤ ✈✐➯♥ ❦❤♦❛ t♦→♥ ❝â ❜❛ ❝→❝❤ ①→❝ ✤à♥❤ ♠ët t➟♣ ♠ð tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝✿ ✣à♥❤ ♥❣❤➽❛ ❤➻♥❤ t❤ù❝✱ sû ❞ö♥❣ ❦❤→✐ ♥✐➺♠ ❜✐➯♥✱ ✈➔ t➟♣ ♠ð ✤÷đ❝ t❤➸ ❤✐➺♥ ❜➡♥❣ ❤đ♣ ❝→❝ q✉↔ ❝➛✉ ♠ð✳ ❈→❝ q✉❛♥ ♥✐➺♠ ♥➔② ð s✐♥❤ ✈✐➯♥ ❦❤→ ❝❤➯♥❤ ❧➺❝❤ s♦ ✈ỵ✐ ✤à♥❤ ♥❣❤➽❛ ❝❤➼♥❤ t❤ù❝✳ ◆❣✉②➵♥ ❚❤à ◆❣å❝ ❆♥❤ ✷ ❈❛♦ ❤å❝ ❑❤â❛ ✷✽ ❚♦→♥ ✷✳ ❚❾P ▼Ð✱ P❍❺◆ ❚❘❖◆● ❚❘❖◆● ❈⑩❈ ❑❍➷◆● ●■❆◆ ✷ ❚➟♣ ♠ð✱ ♣❤➛♥ tr♦♥❣ tr♦♥❣ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ ✷✳✶ ❚➟♣ ♠ð tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ♠❡tr✐❝ ✣à♥❤ ♥❣❤➽❛ ✷✳✶✳ ❑❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ❧➔ ♠ët ❝➠♣ (X, d) tr♦♥❣ ✤â ❳ ❧➔ t➟♣ ❤ñ♣✱ ❞✿X × X → R ❧➔ ♠ët ❤➔♠ ①→❝ ✤à♥❤ tr➯♥ X × X t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉✿ ✶✳❱ỵ✐ ♠å✐ x, y ∈ X ✿ d(x, y) ≥ 0; d(x, y) = ⇔ x = y ✭❚✐➯♥ ỗ t ợ x, y X : d(x, y) = d(y, x) ✭❚✐➯♥ ✤➲ ✤è✐ ①ù♥❣✮ ✸✳❱ì✐ ♠å✐ x, y, z ∈ X : d(x, z) ≥ d(x, y) + d(y, x) ✭❚✐➯♥ ✤➲ t❛♠ ❣✐→❝✮✳ ❍➔♠ ❞ ✤÷đ❝ ❣å✐ ❧➔ ♠➯tr✐❝ tr➯♥ ❳✳ ▼é✐ ♣❤➛♥ tû ❝õ❛ ❳ ✤÷đ❝ ❣å✐ ❧➔ ♠ët ✤✐➸♠ ❝õ❛ ❦❤ỉ♥❣ ❣✐❛♥ ❳✳ ❙è ❞✭①✱②✮ ✤÷đ❝ ❣å✐ ❧➔ ❦❤♦↔♥❣ ❝→❝❤ ❣✐ú❛ ❤❛✐ ✤✐➸♠ ①✱②✳ ❱➼ ❞ư ✷✳ ❚➟♣ ❤đ♣ ❝→❝ sè t❤ü❝ R ✈➔ t➟♣ ❤ñ♣ ❝→❝ sè ♣❤ù❝ C ❧➔ ♥❤ú♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝✱ ✈ỵ✐ ♠➯tr✐❝✿ d(x, y) = |x − y|✱ ✈ỵ✐ ♠å✐ x, y ∈ R✭❤♦➦❝ C✮ ✣à♥❤ ♥❣❤➽❛ ✷✳✷✳ ●✐↔ sû (X, d) ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ xo ∈ X ✈➔ r ❧➔ ♠ët sè ❞÷ì♥❣✳ ❚➟♣ ❤ñ♣ S(xo , r) = {x ∈ X| d(x, xo ) < r} ✤÷đ❝ ❣å✐ ❧➔ ❤➻♥❤ ❝➛✉ ♠ð t➙♠ xo ❜→♥ ❦➼♥❤ r✳ ❱➼ ❞ö ✸✳ ▲➜② X = R ợ tr tổ tữớ d(x, y) = |x − y| ✱ ✈ỵ✐ ♠å✐ x, y ∈ R ❍➻♥❤ ❝➛✉ ♠ð t➙♠ ✶ ❜→♥ ❦➼♥❤ ✷ ❧➔ ❣➻❄ ❚r↔ ❧í✐✿ B(1, 2) = x ∈ R : d(x, 1) = |x − 1| < = x ∈ R : −1 < x < = (−1, 3) ❚ø ✤â s✉② r❛ B(1; 2) = (−1, 3) ❚ê♥❣ q✉→t tr♦♥❣ R t❛ ❝â ❤➻♥❤ ❝➛✉ ♠ð ❧➔ ♠ët ❦❤♦↔♥❣✳ ❚➟♣ ❤ñ♣ S[xo , r] = {x ∈ X|d(x, xo ) ≤ r} ✤÷đ❝ ❣å✐ ❧➔ ❤➻♥❤ ❝➛✉ ✤â♥❣ t➙♠ xo ❜→♥ ❦➼♥❤ r✳ ❱ỵ✐ ❆✱❇ ❧➔ ✷ t➟♣ ❝♦♥ ❦❤→❝ ré♥❣ tr♦♥❣ ❳✱ t❛ ❣å✐✿ ◆❣✉②➵♥ ❚❤à ◆❣å❝ ❆♥❤ ✸ ❈❛♦ ❤å❝ ❑❤â❛ ✷✽ ❚♦→♥ ✷✳ ❚❾P ▼Ð✱ P❍❺◆ ❚❘❖◆● ❚❘❖◆● ❈⑩❈ ❑❍➷◆● ●■❆◆ d(A, B) = infx∈A;y∈B d(x, y) ❧➔ ❦❤♦↔♥❣ ❝→❝❤ ❝õ❛ ❤❛✐ t➟♣ ❝♦♥ ❆✱ ❇✳ ✣à♥❤ ♥❣❤➽❛ ✷✳✸✳ ✣✐♥❤ ♥❣❤➽❛ t➟♣ ♠ð t❤❡♦ ❤➻♥❤ r ổ tr tũ ỵ ❝➛✉ ♠ð ❧➔ t➟♣ ♠ð✳ ❚ù❝ ❧➔ t➟♣ ❝♦♥ A X ữủ t ợ x A t tỗ t > s ❝❤♦ B(x, ε) ⊂ A ❚❤➟t ✈➟②✿ ●✐↔ sû B(a, r) ❧➔ ❤➻♥❤ ❝➛✉ ♠ð t➙♠ ❛ ❜→♥❤ ❦➼♥❤ r tr♦♥❣ ❳✳ ❑❤✐ ✤â ✈ỵ✐ ♠å✐ x ∈ B(a, r) t❛ ❝â d(x, a) < r✳ ✣➦t ε = r − d(x, y) > ❳➨t ❤➻♥❤ ❝➛✉ ♠ð B(x, ε)✳ ❚❛ ❝❤ù♥❣ ♠✐♥❤ B(x, ε) ⊂ B(a, r)✳ ◆➳✉ y ∈ B(x, ε) t❤➻ d(x, y) < ε✳ ❑❤✐ ✤â d(y, a) ≤ d(x, y) = d(x, a) < ε + d(x, a) = r ♥➯♥ y ∈ B(a, r) ❉♦ ✤â B(x, ε) ⊂ B(a, r)✳ ❱➟② B(a, r) ❧➔ t➟♣ ♠ð✳ ✣à♥❤ ♥❣❤➽❛ ✷✳✹✳ ●✐↔ sû ❆ ❧➔ ♠ët t➟♣ ❝♦♥ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ (x, d)✳ ✣✐➸♠ xo ❝õ❛ ❳ ✤÷đ❝ ❣å✐ ❧➔ ✤✐➸♠ tr♦♥❣ ❝õ❛ t ủ tỗ t ởt S(xo , r) ⊂ A✳ ❚➟♣ t➜t ❝↔ ❝→❝ ✤✐➸♠ tr♦♥❣ ❝õ❛ t➟♣ ❤đ♣ ❆ ✤÷đ❝ ❣å✐ ❧➔ ♣❤➛♥ tr♦♥❣ t ủ ỵ A0 P❤➛♥ tr♦♥❣ ❝õ❛ ♠ët t➟♣ ❤ñ♣ ❝â t❤➸ ❧➔ t➟♣ ré♥❣✳ ✣à♥❤ ♥❣❤➽❛ ✷✳✺✳ ✣à♥❤ ♥❣❤➽❛ t➟♣ ♠ð t❤❡♦ ✤✐➸♠ tr♦♥❣ ▼ët t➟♣ ❤đ♣ G ⊂ X ✤÷đ❝ ❣å✐ ❧➔ ♠ð ♥➳✉ ♠å✐ ✤✐➸♠ ❝õ❛ ● ✤➲✉ ❧➔ ✤✐➸♠ tr♦♥❣ ❝õ❛ ♥â✳ ❍✐➸♥ ♥❤✐➯♥ t➟♣ X ✈➔ φ ✤➲✉ ❧➔ ❝→❝ t➟♣ ♠ð tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ✭❳✱❞✮✳ ▼é✐ ❤➻♥❤ ❝➛✉ ♠ð ❧➔ t➟♣ ♠ð tr♦♥❣ (X, d)✳ ❱➼ ❞ö ✹✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣✿ (a, b) ⊂ R ❧➔ t➟♣ ♠ð tr♦♥❣ R✳ ❈❤ù♥❣ ♠✐♥❤✳ ❱ỵ✐ ✤✐➸♠ ❜➜t ❦ý x0 ∈ (a, b) t❛ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ x0 ❧➔ ✤✐➸♠ tr♦♥❣ ❝õ❛ (a, b)✳ ❚❤➟t ✈➙②✿ ▲➜② < r < min{x0 a; b x0 } t tỗ t↕✐ B(x0 , r) ⊂ (a, b)✳ ❱ỵ✐ ♠å✐ ✤✐➸♠ x ∈ B(x0 , r) =⇒ d(x0 , x) < r✳ =⇒ |x0 − x| < r ⇐⇒ −r < x0 − x < r ⇐⇒ −r + x0 < x < x0 + r✳ ❑➳t ❤đ♣ ✈ỵ✐ r < x0 − a ✈➔ r < b − x0 ❤❛② −r + x0 > a ✈➔ r + x0 < b ◆❣✉②➵♥ ❚❤à ◆❣å❝ ❆♥❤ ✹ ❈❛♦ ❤å❝ ❑❤â❛ ✷✽ ❚♦→♥ ✷✳ ❚❾P ▼Ð✱ P❍❺◆ ❚❘❖◆● ❚❘❖◆● ❈⑩❈ ❑❍➷◆● ●■❆◆ ❚ø ✤â s✉② r❛ a < x < b ❦➨♦ t❤❡♦ x ∈ (a, b)✳ ❉♦ ✤â B(x0 , r) ⊂ (a, b)✳ ❱➟② x0 ❧➔ ✤✐➸♠ tr♦♥❣ ❝õ❛ (a, b)✳ ❚➼♥❤ ❝❤➜t ✷✳✻✳ ❍å t➜t ❝↔ ❝→❝ t➟♣ ❝♦♥ ♠ð tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ♠❡tr✐❝ (X, d) ❝â ❝→❝ t➼♥❤ ❝❤➜t s❛✉ ✿ ❛✳ φ ✈➔ X ❧➔ ❝→❝ t➟♣ ủ ủ ởt tũ ỵ t➟♣ ♠ð ❧➔ ♠ët t➟♣ ♠ð✳ ❝✳ ●✐❛♦ ❝õ❛ ♠ët ❤å ❤ú✉ ❤↕♥ ❝→❝ t➟♣ ♠ð ❧➔ t➟♣ ♠ð✳ ❈❤ù♥❣ ♠✐♥❤✳ ❜✳ ●✐↔ sû Ut{ t ∈ T }✱ ❧➔ ởt tũ ỵ ỳ t tr ổ ❣✐❛♥ ♠➯tr✐❝ (X, d)✳ ❚❛ ❝❤ù♥❣ ♠✐♥❤ U = t∈T Ut ❧➔ ♠ët t➟♣ ♠ð✳ ❚❤➟t ✈➟②✿ ●✐↔ sû X U tũ ỵ õ x U1 ợ ∈ t ♥➔♦ ✤â✳ ❱➻ U ✱ ♠ð ♥➯♥ tỗ t ởt S(x, r) U1 ❞♦ ✤â S(x, r) ⊂ U ✳ ❱➟② U ❧➔ t➟♣ ♠ð✳ ❝✳ ●✐↔ sû U1 , U2 , , Un ❧➔ ♥❤ú♥❣ t➟♣ ♠ð✳ ❚❛ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ V = n i=1 Ui ❧➔ t➟♣ ♠ð✳ ❚❤➟t ✈➟②✿ ◆➳✉ x ∈ V t❤➻ x ∈ U ✳ ❱ỵ✐ ♠å✐ i = 1, , n✱ ✈➻ ♠é✐ Ui ♠ð ♥➯♥ tỗ t ởt số ữỡ r s S(x, ri ) ⊂ Ui ✱ ✈ỵ✐ i = 1, n ✣➦t r = minr1 , r2 , , ru ❑❤✐ ✤â ❤✐➸♥ ♥❤✐➯♥ S(x, r) ⊂ Ui ✈ỵ✐ i = 1, 2, , n✳ ❉♦ ✤â S(x, r) ⊂ V ✳ ❱➙② V ❧➔ ♠ët t➟♣ ♠ð✳ ✣à♥❤ ❧➼ ✷✳✼✳ ợ x (X, d) tũ ỵ t t ❦ý U ⊃ X ❝❤ù❛ ✤✐➸♠ X ✤÷đ❝ ❣å✐ ❧➔ ❧➙♥ ❝➟♥ ❝õ❛ ✤✐➸♠ X ♥➳✉ U ❝❤ù❛ ♠ët t➟♣ ♠ð ❝❤ù❛ ①✳ ❍✐➸♥ ♥❤✐➯♥✱ t➟♣ ❆ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ X ❧➔ ♠ð ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ✈ỵ✐ ♠é✐ x A ổ tỗ t ởt U ❝õ❛ X ❝❤ù❛ tr♦♥❣ A ✈➔ ❤✐➸♥ ♥❤✐➯♥ t❛ ❧✉æ♥ ❝â✿ ✶✳ A0 ❧➔ t➟♣ ♠ð✱ ✈➔ ✤â ❧➔ t➟♣ ♠ð ❧ỵ♥ ♥❤➜t ❝❤ù❛ tr♦♥❣ ❆✳ ✷✳ ❚➟♣ A ❧➔ ♠ð ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ A = A0 ✳ ✸✳ ◆➳✉ A ⊂ B t❤➻ A0 ⊂ B ✳ ◆❣✉②➵♥ ❚❤à ◆❣å❝ ❆♥❤ ✺ ❈❛♦ ❤å❝ ❑❤â❛ ✷✽ ❚♦→♥ ✷✳ ❚❾P ▼Ð✱ P❍❺◆ ❚❘❖◆● ❚❘❖◆● ❈⑩❈ ❑❍➷◆● ●■❆◆ ✷✳✷ ❚➟♣ ♠ð✱ ♣❤➛♥ tr♦♥❣ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❚♦♣♦ ❚r♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❚♦♣♦ ❦❤→✐ ♥✐➺♠ t➟♣ ♠ð ❧➔ ❦❤→✐ ♥✐➺♠ ❝ì sð✳ ✣à♥❤ ♥❣❤➽❛ ✷✳✽✳ ❈❤♦ t➟♣ ❤ñ♣ X ✳ ❍å τ ♥❤ú♥❣ t➟♣ ❝♦♥ ❝õ❛ X ✤÷đ❝ ❣å✐ ❧➔ ♠ët t♦♣♦ tr➯♥ X ♥➳✉ t❤ä❛ ♠➣♥ ✸ t✐➯♥ ✤➲ s❛✉✿ (T1 ) φ, X ∈ τ (T2 ) ❍ñ♣ ❝õ❛ tũ ỵ tỷ tở tû t❤✉ë❝ τ (T3 ) ●✐❛♦ ❝õ❛ ❤❛✐ ♣❤➛♥ tû ❜➜t ❦➻ t❤✉ë❝ τ ❧➔ ♣❤➛♥ tû t❤✉ë❝ τ ❑❤✐ ✤â ❝➦♣ (X, τ ) ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ t♦♣♦ ✈➔ ❝á♥ ✤÷đ❝ ✈✐➳t ❣å♥ ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ t♦♣♦ X ✳ ❈→❝ ♣❤➛♥ tû ∈ τ ✤÷đ❝ ❣å✐ ❧➔ ♠ët t➟♣ ♠ð tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ t♦♣♦ X ❱➼ ❞ư ✺✳ ✣➦t X = [0, 5], A = [0, 2] ❳➨t τ = φ, X ❧➔ ♠æt t♦♣♦ t❤æ tr➯♥ ❳ ❉♦ A = φ, A = X =⇒ A ∈ /τ =⇒ A ❦❤æ♥❣ ♣❤↔✐ ❧➔ t➟♣ ♠ð tr♦♥❣ (X, τ ) ❚✉② ♥❤✐➯♥ ①➨t τ = (P)(X) ✿ ❚➟♣ t➜t ❝↔ ❝→❝ t➟♣ ❝♦♥ ❝õ❛ ❳ ❉♦ A = [0, 2] ⊂ [0, 5] = X =⇒ A ∈ τ =⇒ ❆ ❧➔ t➟♣ ♠ð tr♦♥❣ (X, τ ) ✣à♥❤ ♥❣❤➽❛ ✷✳✾✳ P❤➛♥ tr♦♥❣ P❤➛♥ tr♦♥❣ ❝õ❛ ♠ët t➟♣ ❝♦♥ A ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ t♦♣♦ ❳ ❧➔ ♠ët t➟♣ ủ ỗ A ổ tở ❜✐➯♥ ❝õ❛ A✳ ❍❛② ♥â✐ rã ❤ì♥✿ ① ❧➔ ♠ët ✤✐➸♠ ♣❤➛♥ tr♦♥❣ ❝õ❛ ❆ ♥➳✉ ① ❝❤ù❛ ✤÷đ❝ ♠ët t➟♣ ❝♦♥ ♠ð ❝õ❛ A ✭❤♦➦❝ ① ❧➔ ♠ët ✤✐➸♠ ♣❤➛♥ tr♦♥❣ ❝õ❛ A ♥➳✉ ❝â ♠ët ❧➙♥ ❝➟♥ ❝õ❛ ① ❝❤ù❛ tr♦♥❣ A ✮ P❤➛♥ tr♦♥❣ ❝õ❛ A ❧➔ ũ õ ũ A ỵ int(A)✱ Int(A) ❤♦➦❝ A0 ✳ ❱➼ ❞ư ✻✳ ❚r♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ tỉ♣♦ t❤ỉ (R, τ ) ✈ỵ✐ ❜➜t ❦ý ❆ ❧➔ t➟♣ ❝♦♥ t❤ü❝ sü ❝õ❛ R t❛ ❝â A0 = φ✳ ❚r♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ t♦♣♦ rí✐ r↕❝ (R, τD )✱ ✈ỵ✐ ❜➜t ❦ý A = (a, b) ⊂ R t❛ ❝â A0 = (a, b)✳ ❚r♦♥❣ ❦❤æ♥❣ ❣✐❛♥ t♦♣♦ t❤ỉ (R, τ ) ✈ỵ✐ τ ❧➔ t♦♣♦ tü ♥❤✐➯♥✱ ❝❤♦ A = (a, b)✳ ❑❤✐ ✤â✱ ♠å✐ ✤✐➸♠ t❤✉ë❝ (a, b) ✤➲✉ ❧➔ ✤✐➸♠ tr♦♥❣ ❝õ❛ A✿ A0 = (a, b)✱ ❝→❝ ✤✐➸♠ ❛✱❜ ❧➔ ✤✐➸♠ ❜✐➯♥ ❝õ❛ A✳ ◆❣✉②➵♥ ❚❤à ◆❣å❝ ❆♥❤ ✻ ❈❛♦ ❤å❝ ❑❤â❛ ✷✽ ❚♦→♥ ✷✳ ❚❾P ▼Ð✱ P❍❺◆ ❚❘❖◆● ❚❘❖◆● ❈⑩❈ ❑❍➷◆● ●■❆◆ ❱➼ ❞ư ✼✳ ❛✳ ❚r♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ t♦♣♦ R ❝❤✉➞♥ ✿ int([0, 1]) = (0, 1)✳ ❜✳ ❚r♦♥❣ ❦❤æ♥❣ ❣✐❛♥ t♦♣♦ R ❝❤✉➞♥✱ ♣❤➛♥ tr♦♥❣ ❝õ❛ t➟♣ Q ❧➔ ré♥❣ ✈➻ ♥â ❦❤æ♥❣ ❝â t➟♣ ❝♦♥ ♠ð ❦❤→❝ ré♥❣ ♥➔♦✱ ♠å✐ ❧➙♥ ❝➟♥ ❝õ❛ ♠ët sè ❤ú✉ t✛ ✤➲✉ ❝❤ù❛ sè ✈æ t✛✳ ❝✳ ❚r♦♥❣ ❦❤æ♥❣ ❣✐❛♥ t♦♣♦ Rn ❝❤✉➞♥✱ ♣❤➛♥ tr♦♥❣ ❝õ❛ ❜➜t ❦ý t➟♣ ❤ú✉ ❤↕♥ ♥➔♦ ✤➲✉ ré♥❣✳ ❞✳ ❚r♦♥❣ ♠å✐ ❦❤ỉ♥❣ ❣✐❛♥ t♦♣♦ t➛♠ t❤÷í♥❣ ✭▲➔ t♦♣♦ ♠➔ ❝❤➾ ❝â ❝→❝ t➟♣ ♠ð ❧➔ t➟♣ ré♥❣ ✈➔ t♦➔♥ ❜ë ❦❤ỉ♥❣ ❣✐❛♥✮ ❳✳ ❈❤ó♥❣ t❛ ❝â int(X) = X ❚➼♥❤ ❝❤➜t ✷✳✶✵✳ ❈❤♦ ❦❤æ♥❣ ❣✐❛♥ t♦♣♦ (X, τ ) ❛✳ ✣è✐ ✈ỵ✐ ❜➜t ❦➻ A ⊂ X t❛ ❝â✿ X = A0 ∪ b(A) ∪ extA ❀ extA = (X A)0 ❈→❝ t➟♣ A0 , extA ❧➔ ♠ð✱ t➟♣ b(A) ❧➔ t➟♣ ✤â♥❣✳ ❜✳❚➟♣ A0 ❧➔ t➟♣ ♠ð ❧ỵ♥ ♥❤➜t tr♦♥❣ A✳ ❝✳❚➙♣ A ❧➔ ♠ð ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ A = A0 ✳ ❞✳ ◆➳✉ B ⊂ A ⊂ X t❤➻ B ⊂ A0 ❡✳ ❱ỵ✐ ♠å✐ A, B ∈ / X t❛ ❝â A0 ∩ B = (A ∩ B)0 ✳ ❈❤ù♥❣ ♠✐♥❤✳ ❛✳ ❍✐➸♥ ♥❤✐➯♥ X = A0 ∪ b(A) ∩ ext(A)✳ ❚❛ õ x extA tỗ t ♠ët ❧➙♥ ❝➟♥ U ❝õ❛ x s❛♦ ❝❤♦ U ⊂ X A✳ ✣✐➲✉ ♥➔② ❞➝♥ ✤➳♥ x ❧➔ ✤✐➸♠ tr♦♥❣ ❝õ❛ X A ❤❛② x ∈ (X A)0 ✳ ❱➟② extA = (X A)0 ❜✳ ❣✐↔ sû ❱ ❧➔ t➟♣ ♠ð ❜➜t ❦ý tr♦♥❣ ❆✳ ❑❤✐ ✤â ❱ ❧➔ ❧➙♥ ❝➟♥ ❝õ❛ ♠å✐ ✤✐➸♠ t❤✉ë❝ ♥â✱ ♥❣❤➽❛ ❧➔✿ ❱ỵ✐ ♠å✐ x ∈ V t❛ ❝â x ∈ V ⊂ A s✉② r❛ ▼å✐ x ∈ V ✤➲✉ ❧➔ ✤✐➸♠ tr♦♥❣ ❝õ❛ ❆✳ ❱➟② V ⊂ A0 ✱ ❤❛② A0 ❧➔ t ợ t tr A ỷ ỵ ❛ ✈➔ ❜ t❛ ❝â ♥❣❛② t➟♣ A ❧➔ t➟♣ ♠ð ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ A = A0 ✳ ❞✳ ●✐↔ sû B ⊂ A ⊂ X ✳ ❱➻ B ⊂ B ⊂ A ❧➔ t➟♣ ♠ð ❧ỵ♥ ♥❤➜t tr♦♥❣ A ♥➯♥ B ⊂ A0 ✳ ❡✳ ●✐↔ sû A, B ⊂ X ✳ ❱➻ A0 ∩ B ❧➔ t➟♣ ♠ð tr♦♥❣ A ∩ B ♥➯♥ t❤❡♦ þ ❜ t❛ ❝â A0 ∩ B ⊂ (A B)0 ữủ ợ t ý X (A B)0 ổ tỗ t U ❝õ❛ x s❛♦ ❝❤♦ U ⊂ A ∩ B ❙✉② r❛ U ⊂ A ✈➔ U ⊂ B ◆❣✉②➵♥ ❚❤à ◆❣å❝ ❆♥❤ ✼ ❈❛♦ ❤å❝ ❑❤â❛ ✷✽ ❚♦→♥ ✸✳ ▼❐❚ ❙➮ Ù◆● ❉Ö◆● ❈Õ❆ ❚❹P ▼Ð ❉♦ ✤â x ∈ A0 ✈➔ x ∈ B ✳ ❱➟② A0 ∩ B = (A ∩ B)0 ✷✳✸ ❚➟♣ ♠ð tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❊✉❝❧✐❞❡ ❑❤ỉ♥❣ ❣✐❛♥ ❊✉❝❧✐❞❡ Rn ❝ơ♥❣ ❧➔ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝✱ ♥â ❝ơ♥❣ ❧➔ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ t♦♣♦ ✈ỵ✐ t♦♣♦ tü ♥❤✐➯♥ s✐♥❤ ❜ð✐ ♠❡tr✐❝✳ ❚♦♣♦ tr➯♥ E n ✤÷đ❝ ❣å✐ ❧➔ ✳ ▼ët t➟♣ ❧➔ t➟♣ ♠ð tr♦♥❣ t♦♣♦ ❊✉❝❧✐❞❡ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ ♥â ❝❤ù❛ ❤➻♥❤ ❝➛✉ ♠ð ❜❛♦ q✉❛♥❤ ♠é✐ ✤✐➸♠ ❝õ❛ ♥â✳ ❚♦♣♦ tữỡ ữỡ ợ ởt t t tr Rn ữ ❧➔ t➼❝❤ ❝õ❛ ♥ ❜↔♥ s❛♦ ❝õ❛ ✤÷í♥❣ t❤➥♥❣ t❤ü❝ ❘✭ ❱ỵ✐ t♦♣♦ ❝❤➼♥❤ t➢❝✮✳ t♦♣♦ ❊✉❝❧✐❞❡ ✸ ▼ët sè ù♥❣ ❞ö♥❣ ❝õ❛ t➙♣ ♠ð ✸✳✶ ❚r♦♥❣ t♦♣♦ ❈→❝ t➟♣ ♠ð✱♣❤➛♥ tr♦♥❣ ❝â ✈❛✐ trá q✉❛♥ trå♥❣ tr♦♥❣ ❚♦♣♦✳ ▼å✐ t➟♣ ❝♦♥ ❆ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ t♦♣♦ X ❝❤ù❛ ➼t ♥❤➜t ♠ët t➟♣ ♠ð ✭❈â t❤➸ ❧➔ t➟♣ ré♥❣❀ t➟♣ ợ t tr ú ữủ tr♦♥❣ ❝õ❛ ❆✮✳ ❚➟♣ ❝♦♥ ♥➔② ❝â t❤➸ ✤÷đ❝ ①➙② ❞ü♥❣ ❜➡♥❣ ❝→❝❤ ❤ñ♣ t➜t ❝↔ ❝→❝ t➟♣ ♠ð ❝❤ù❛ tr♦♥❣ A✳ ✣à♥❤ ♥❣❤➽❛ ✸✳✶✳ ❈❤♦ X ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ t♦♣♦✱ t➟♣ ❝♦♥ E ❝õ❛ X ✤❣❧ t➟♣ ✤â♥❣ ♥➳✉ X E ❧➔ t➟♣ ♠ð✳ ❚❛ t❤➜② ✤➸ ❝❤ù♥❣ ♠✐♥❤ ♠ët t➟♣ ❧➔ t➟♣ ✤â♥❣ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ t♦♣♦ t❤➻ ❝❤ó♥❣ t❛ ❝ơ♥❣ sû ❞ư♥❣ t➟♣ ♠ð✳ ❚ø t➟♣ ♠ð ✤➸ s✉② r❛ ✤÷đ❝ t➟♣ ✤â♥❣✳ ❱➼ ❞ư ✽✳ ❚r♦♥❣ ❦❤æ♥❣ ❣✐❛♥ t♦♣♦ X ❝❤♦ ❤❛✐ t➟♣ A ⊂ X ✱ B ⊂ X ✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣✿ ◆➳✉ A ✈➔ B ✤➲✉ ❧➔ t➟♣ ✤â♥❣ t❤➻ A ∪ B ❧➔ t➟♣ ✤â♥❣❄ ❈❤ù♥❣ ♠✐♥❤✳ ✣➸ ❝❤ù♥❣ ♠✐♥❤ A ∪ B ❧➔ t➟♣ ✤â♥❣ t❤➻ ❝❤ó♥❣ t❛ ✤✐ ❝❤ù♥❣ ♠✐♥❤ X (A ∪ B) ❧➔ t➟♣ ♠ð✳ ❚❤➟t ✈➙②✿ ❚❛ ❝â A, B ❧➔ ❝→❝ t➟♣ ✤â♥❣✳ ❉♦ ✤â X A ✈➔ X B ❧➔ ❝→❝ t➟♣ ♠ð tr♦♥❣ X✳ ▼➦t ❦❤→❝ t❛ ❝â X ❧➔ ❦❤æ♥❣ ❣✐❛♥ t♦♣♦ ♥➯♥ ❣✐❛♦ ❤ú✉ ❤↕♥ ❝→❝ t➟♣ ♠ð ❧➔ ❝→❝ t➟♣ ♠ð✳ ❱➻ t❤➳✿ ◆❣✉②➵♥ ❚❤à ◆❣å❝ ❆♥❤ ✽ ❈❛♦ ❤å❝ ❑❤â❛ ✷✽ ❚♦→♥ ✸✳ ▼❐❚ ❙➮ Ù◆● ❉Ö◆● ❈Õ❆ ❚❹P ▼Ð (X A) ∩ (X B) ❧➔ ✶ t➟♣ ♠ð tr♦♥❣ ❳ ❤❛② X (A ∪ B) ❧➔ ♠ët t➟♣ ♠ð tr♦♥❣ ❳✳ ✣à♥❤ ♥❣❤➽❛ ✸✳✷✳ ❈❤♦ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ t♦♣♦ (X, τ ) ✈➔ (Y, σ)✳ ⑩♥❤ ①↕ f : X −→ Y tø ❦❤æ♥❣ ❣✐❛♥ t♦♣♦ (X, τ ) ✈➔♦ ❦❤æ♥❣ ❣✐❛♥ t♦♣♦ (Y, σ)✳ ⑩♥❤ ①↕ f ✤÷đ❝ ❣å✐ ❧➔ ❧✐➯♥ tư❝ t↕✐ ✤✐➸♠ x ∈ X ♥➳✉ ✈ỵ✐ ♠é✐ ❧➙♥ ❝➟♥ ❱ ❝õ❛ ✤✐➸♠ f (x) Y tỗ t U x s ❝❤♦ f (U ) ⊂ V ✳ ⑩♥❤ ①↕ f ✤÷đ❝ ❣å✐ ❧➔ ❧✐➯♥ tư❝ tr➯♥ X ♥➳✉ f ❧✐➯♥ tö❝ t↕✐ ♠å✐ ✤✐➸♠ x ∈ X ❱✐➺❝ ①➨t t➼♥❤ ♠ð ❝õ❛ ❝→❝ t➟♣ ❤đ♣ ❧➔ ❝ỉ♥❣ ❝ư ❣✐ó♣ ❝❤ó♥❣ t❛ ❝â t❤➸ ①➨t ✤÷đ❝ t➼♥❤ ❧✐➯♥ tư❝ ❝õ❛ →♥❤ ①↕ tø ❦❤æ♥❣ ❣✐❛♥ t♦♣♦ ♥➔② ✈➔♦ ❦❤æ♥❣ ❣✐❛♥ t♦♣♦ ❦❤→❝✳ ❚➼♥❤ ❝❤➜t ✸✳✸✳ ❈❤♦ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ t♦♣♦ (X, τ ) ✈➔ (Y, σ)✳ ⑩♥❤ ①↕ f : X −→ Y tø ❦❤æ♥❣ ❣✐❛♥ t♦♣♦ (X, τ ) ✈➔♦ ❦❤ỉ♥❣ ❣✐❛♥ t♦♣♦ (Y, σ)✳ ⑩♥❤ ①↕ ❢ ♥➔② ✤÷đ❝ ❣å✐ ❧➔ ❧✐➯♥ tö❝ ♥➳✉ t↕♦ ↔♥❤ ❝õ❛ ♠å✐ t➟♣ ♠ð tr♦♥❣ Y ❧➔ t➟♣ ♠ð tr♦♥❣ X ✳ ❱➔ →♥❤ ①↕ ❢ ✤÷đ❝ ❣å✐ ❧➔ ♠ð ♥➳✉ ↔♥❤ ❝õ❛ ♠å✐ t➟♣ ♠ð tr♦♥❣ X ❧➔ ♠ð tr♦♥❣ Y ❈❤ù♥❣ ♠✐♥❤✳ ❚❤➟t ✈➙②✿ ◆❤➢❝ ❧↕✐ U ⊂ X ❧➔ t➟♣ ♠ìt ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ U ❧➔ ❧➙♥ ❝➟♥ ❝õ❛ ♠å✐ x ∈ U ✳ ●✐↔ sû ❢ ❧➔ ♠ët →♥❤ ①↕ ❧✐➯♥ tö❝ ✈➔ V ❧➔ t➟♣ ♠ð ❜➜t ❦ý tr♦♥❣ Y ✳ ❱ỵ✐ ♠å✐ ✤✐➸♠ x ∈ f − 1(V ) ✤➸ ❝❤ù♥❣ ♠✐♥❤ f − 1(V ) ❧➔ t➟♣ ♠ð t❛ s➩ ❝❤ù♥❣ ♠✐♥❤ f − 1(V ) ❧➔ ❧➙♥ ❝➟♥ ❝õ❛ ✤✐➸♠ x✳❚❤➟t ✈➟②✿ ❱➻ x ∈ f − 1(V ) ♥➯♥ f (x) ∈ V ✳ ❱➻ V ❧➔ t➟♣ ♠ð ♥➯♥ V ❧➔ ❧➙♥ ❝➟♥ ❝õ❛ f (x)✳ ❱➻ f ❧➔ →♥❤ ①↕ ❧✐➯♥ tử tỗ t U x s ❝❤♦ f (U ) ⊂ V ✳ =⇒ U ⊂ f − 1(V ) ❚❤❡♦ t➼♥❤ ❝❤➜t ❝õ❛ ❧➙♥ ❝➟♥ t❤➻ t❛ ❝ô♥❣ ❝â f − 1(V ) ❝ô♥❣ ❧➔ ❧➙♥ ❝➟♥ ❝õ❛ x✳ ❱➟② f − 1(V ) ❧➔ ♠ët t➟♣ ♠ð ✭❞♣❝♠✮✳ ▼é✐ t➟♣ ♠ð ❜➜t ❦ý tr➯♥ ✤÷í♥❣ t❤➥♥❣ t❤ü❝ ❧➔ ❤đ♣ ❝õ❛ ✤➳♠ ✤÷đ❝ ❝→❝ ❦❤♦↔♥❣ ♠ð ◆❣✉②➵♥ ❚❤à ◆❣å❝ ❆♥❤ ✾ ❈❛♦ ❤å❝ ❑❤â❛ ✷✽ ❚♦→♥ ✸✳ ▼❐❚ ❙➮ Ù◆● ❉Ö◆● ❈Õ❆ ❚❹P ▼Ð ✸✳✷ ❚r♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ❈→❝ t➟♣ ♠ð ❝â ✈❛✐ trá q✉❛♥ trå♥❣✱ tø t➟♣ ♠ð ❝❤ó♥❣ t❛ ❝â t❤➸ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ ❝→❝ ❦❤→✐ ♥✐➺♠ ❧✐➯♥ q✉❛♥ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝✳ ❈❤➥♥❣ ❤↕♥ ♥❤÷ tr♦♥❣ ✈✐➺❝ ❝❤ù♥❣ ♠✐♥❤ ♠ët t➟♣ ❧➔ t➟♣ ✤â♥❣✳ ✣à♥❤ ♥❣❤➽❛ ✸✳✹✳ ❚➟♣ ❝♦♥ F ❝õ❛ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ X ✤÷đ❝ ❣å✐ ❧➔ t➟♣ ✤â♥❣ ♥➳✉ X F ❧➔ t➟♣ ♠ð✳ ❱➼ ❞ö ✾✳ ✣♦↕♥ [a, b] ⊂ R ❧➔ ♠ët t➟♣ ✤â♥❣ ✈➻ R [a, b] = (−∞, a) ∪ (b, +∞) ❧➔ t➟♣ ♠ð✳ ❱➼ ❞ö ✶✵✳ ❈❤ù♥❣ ♠✐♥❤ A = {(x1, x2) ∈ R2 : −x1 + 3x2 > 0} ❧➔ ♠ët t➟♣ ♠ð ❈❤ù♥❣ ♠✐♥❤✳ ❳➨t →♥❤ ①↕ f : R2 −→ R, (x1 , x2 ) −→ −x1 + 3x2 ❚❛ t❤➜② ❢ ❧➔ →♥❤ ①↕ ❧✐➯♥ tö❝ tr➯♥ R2 ✳ ▲➜② ✤÷đ❝ ❜➜t ❦➻ (x1 , x 2) ∈ R2 t❛ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ f ❧✐➯♥ tö❝ t↕✐ ✤✐➸♠ (x1 , x 2) tù❝ t❛ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤✿ ❱ỵ✐ ♠å✐ > tỗ t > s d((x1 , x2 ), (x1 , x2 )) < γ t❤➻ |f (x1 , x2 ) − f (x1 , x2 )| < ε✱ ✈ỵ✐ ♠å✐ (x1 , x2 ) ∈ R2 ✳ ❚❛ ❝â✿ |f (x1 , x2 )−f (x1 , x2 )| = |−x1 +3x2 +x1 −3x2 | = |(−x1+x1 )+3(x2 −x2 )| ⑩♣ ❞ö♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ❇✉♥❤✐❛❝♦♣s❦✐ ❝❤♦ ❜ë sè 1, 3, −(x1+x1 ), (x2 −x2 )✳ ❚❛ ❝â✿ |f (x1 , x2 ) − f (x1 , x2 )| ≤ (12√+ 32 ).[(−x1 + x1 )2 + (x2 − x2 )2 ] ⇐⇒ |f (x1 , x2 ) − f (x1 , x2 )| ≤ 10 (−x1 + x1 )2 + (x2 − X2 )2 = 10 ợ > 0, tỗ t↕✐ γ = √ε10 ♠➔✿ d((x1 , x2 ), (x1 , x2 )) < γ ⇐⇒ |f (x1 , x2 ) − f (x1 , x2 )| < ε✳ ❱➟② f ❧✐➯♥ tö❝ tr➯♥ R2 f − 1(0, +∞) = {(x1 , x2 ) ∈ R2 |f (x1 , x2 ) ∈ (0, +∞)} = {−x1 + 3x2 ∈ (0, +∞)} ❂{(x1 , x2 ) ∈ R2 | − x1 + 3x2 > 0} = A ❚❛ ❝â (0; +∞) ❧➔ t➟♣ ♠ð tr♦♥❣ R ✈➻ ✈ỵ✐ ♠å✐ x0 ∈ (0; +) t ổ tỗ t B(x0 , r) ⊂ (0; +∞) ◆❣✉②➵♥ ❚❤à ◆❣å❝ ❆♥❤ ✶✵ ❈❛♦ ❤å❝ ❑❤â❛ ✷✽ ❚♦→♥ ✹✳ P❍❹◆ ❚➑❈❍ ❚❘■ ❚❍Ù❈ ▲❯❾◆ ❈Õ❆ ❑❍⑩■ ◆■➏▼ ❚❾P ▼Ð ✸✳✸ ▼ët sè ❧÷✉ þ ❚➼♥❤ ❝❤➜t ♠ð ❝õ❛ ♠ët t➟♣ U tr♦♥❣ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ ♥➔♦ ✤â ❝â t❤➸ ✤÷đ❝ ❜↔♦ t♦➔♥ tr♦♥❣ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ ❧ỵ♥ ❤ì♥✳ ❈❤➥♥❣ ❤↕♥✱ ♥➳✉ U ❧➔ t➟♣ ❤ñ♣ ❝→❝ sè ❤ú✉ t✛ tä♥❣ ❦❤♦↔♥❣ (0, 1)✱ ❦❤✐ ✤â U ❧➔ ♠ð tr♦♥❣ t➟♣ ❝→❝ sè ❤ú✉ t✛✱ ♥❤÷♥❣ ❦❤ỉ♥❣ ♠ð tr♦♥❣ t➟♣ ❝→❝ sè t❤ü❝✳✣â ❧➔ ✈➻✱ ❦❤✐ ①➨t U tr♦♥❣ t➙♣ ❝→❝ sè ❤ú✉ t✛✱ x U ỗ ❝→❝ sè ❤ú✉ t✛✳ ❚✉② ♥❤✐➯♥✱ ❦❤✐ ①➨t U ♥❤÷ t➟♣ ❝♦♥ ❝õ❛ t➟♣ ❝→❝ sè t❤ü❝✱ ❝→❝ ❧➙♥ ❝➟♥ ❜➜t ❦ý ❝õ❛ ✤✐➸♠ x ✤➲✉ ❝❤ù❛ ❝↔ ❝→❝ ✤✐➸♠ ✈æ t✛ ✈➔ ❤ú✉ t✛ ✈➔ ❞♦ ✤â ❦❤æ♥❣ t❤➸ ♥➡♠ trå♥ tr♦♥❣ U ✳ ▼ët sè t➟♣ ❤ñ♣ ✈ø❛ ♠ð✱ ✈ø❛ ❧➔ ✤â♥❣✿ ❚r♦♥❣ R ✈➔ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ ❧✐➯♥ t❤æ♥❣✱ ❝❤➾ ❝â t➟♣ φ ✈➔ t♦➔♥ ❜ë ❦❤æ♥❣ ❣✐❛♥ ❧➔ ✈ø❛ ✤â♥❣ ✈ø❛ ♠ð✳ ❚➟♣ ❤ñ♣ ❝→❝ sè ❤ú✉ t✛ ♥❤ä ❤ì♥ ❈➠♥✭✷✮ ❧➔ ✈ø❛ ❞â♥❣ ✈ø❛ ♠ð tr♦♥❣ ❝→❝ t➟♣ sè ❤ú✉ t✛✳ ❚r♦♥❣ ❦❤✐ ✤â✱ ♠ët sè t➟♣ ❦❤→❝ ❧➔ ❦❤ỉ♥❣ ✤â♥❣ ❝ơ♥❣ ❦❤ỉ♥❣ ♠ð✳ ❱➼ ❞ö (0, 1] tr♦♥❣ R✳ ✹ P❤➙♥ t➼❝❤ tr✐ t❤ù❝ ❧✉➟♥ ❝õ❛ ❦❤→✐ ♥✐➺♠ t➟♣ ♠ð ✹✳✶ ◗✉→ tr➻♥❤ ❤➻♥❤ t❤➔♥❤ ✈➔ ♣❤→t tr✐➸♥ ❝õ❛ ❦❤→✐ ♥✐➺♠ t➟♣ ♠ð tr♦♥❣ ❧à❝❤ sû ✹✳✶✳✶✳ ❉❡❞❡❦✐♥❞ ✈ỵ✐ q✉❛♥ ♥✐➺♠ ♥❣✉②➯♥ t❤õ② ✈➲ t rữợ ữớ õ ỳ ỵ tữ ✤➛✉ t✐➯♥ ✈➲ t➟♣ ♠ð ♠➦❝ ❞ị ỉ♥❣ ❣å✐ ♥â ✈ỵ✐ ♠ët ❝→✐ t➯♥ ❦❤→❝ ❧➔ ✧❑♦r♣❡r✧✳ ◆❣➔② ✶✾✴✵✶✴✶✽✼✾✱ ❉❡❞❡❦✐♥❞ ❝â ❣û✐ ❝❤♦ ❈❛♥t♦r ✶ ❜ù❝ t❤÷✱ tr♦♥❣ ✤â ❝â ✤➲ ❝➟♣ ✤➳♥ ❜↔♥ t❤↔♦ ❝â tü❛ ✤➲ ●❡♥❡r❛❧ ❚❤❡♦r❡♠s ❛❜♦✉t ❙♣❛❝❡s✱ ❜➢t ✤➛✉ ✈ỵ✐ ✤à♥❤ ♥❣❤➽❛ ❝õ❛ ❝→✐ ♠➔ æ♥❣ ❣å✐ ❧➔ ♠ët✧❑♦r♣❡r✧✿ ▼ët ❤➺❬tù❝ ❧➔ t➟♣ ❪ ❝→ ✤✐➸♠ ♣✱ p ✳✳✳ t↕♦ t❤➔♥❤ ♠ët ❑♦r♣❡r ♥➳✉ ✈ỵ✐ ♠é✐ ✤✐➸♠ ♣ ❝õ❛ ♥â✱ ❝â ♠ët ✤ë ❞➔✐ ❞ s❛♦ ❝❤♦ t➜t ❝↔ ❝→❝ ✤✐➸♠ ❝â ❦❤♦↔♥❣ ❝→❝❤ tø ❝❤ó♥❣ ✤➳♥ ♣ ♥❤ä ❤ì♥ ❞ t❤➻ t❤✉ë❝ P✳ ❈→❝ ✤✐➸♠ ♣✱p ✳✳✳❬✤÷đ❝ ❣å✐ ❧➔❪ ♥➡♠ tr♦♥❣ P✳ ◆❤÷ ✈➟②✱ ❑♦r♣❡r ❝õ❛ ❉❡❞❡❦✐♥❞ ❝❤➼♥❤ ①→❝ ❧➔ ♠ët t➟♣ ♠ð tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❊✉❝❧✐❞❡✱ ❝â t❤➸ ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♥ ❝❤✐➲✉✳ ➷♥❣ sû ❞ö♥❣ ❦❤→✐ ♥✐➺♠ ❑♦r♣❡r ✤➸ ✤à♥❤ ♥❣❤➽❛ t❤➳ ♥➔♦ ❧➔ ♠ët ✤✐➸♠ ♥➡♠ tr♦♥❣ P✳ ◆❤÷ ✈➟②✱ q✉❛♥ ♥✐➺♠ ❝õ❛ ❉❡❞❡❦✐♥❞ ✈➲ t➟♣ ♠ð ❧➔ q✉❛♥ ♥✐➺♠ ❤➻♥❤ ❤å❝✱ ◆❣✉②➵♥ ❚❤à ◆❣å❝ ❆♥❤ ✶✶ ❈❛♦ ❤å❝ ❑❤â❛ ✷✽ ❚♦→♥ ✹✳ P❍❹◆ ❚➑❈❍ ❚❘■ ❚❍Ù❈ ▲❯❾◆ ❈Õ❆ ❑❍⑩■ ◆■➏▼ ❚❾P ▼Ð ①❡♠ ①➨t t➟♣ ♠ð t❤❡♦ ❦❤♦↔♥❣ ❝→❝❤ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❊✉❝❧✐❞❡ ♥ ❝❤✐➲✉ ♠➔ ♥❣➔② ♥❛② ❣å✐ ❧➔ ✤à♥❤ ♥❣❤➽❛ t❤❡♦ q✉↔ ❝➛✉ ♠ð✳ ❚➟♣ ♠ð r❛ ✤í✐ ợ trỏ ởt ổ ữủ sỷ ❞ö♥❣ ✤➸ ①➨t t❤➳ ♥➔♦ ❧➔ ♠ët ✤✐➸♠ ♥➡♠ ❜➯♥ ♥❣♦➔✐ ❝õ❛ ♠ët ❑♦r♣❡r ✳ ◗✉❛♥ ♥✐➺♠ ❉❡❞❡❦✐♥❞ ✈➲ t➟♣ ♠ð ❝â ❤➻♥❤ t❤ù❝ ❝õ❛ ♠ët ❦❤→✐ ♥✐➺♠ ❝➟♥ t♦→♥ ❤å❝ ✈➔ ❝â ❝ì ❝❤➳ ❝ỉ♥❣ ❝ư t÷í♥❣ ♠✐♥❤✳ ✹✳✶✳✷✳ P❡❛♥♦ ✈➔ ❏♦r❞❛♥ ✈ỵ✐ t✐➳♣ ❝➟♥ ❦❤ỉ♥❣ t❤➔♥❤ ❝ỉ♥❣ ✈ỵ✐ t tr ổ sỷ ỵ tữ tê♥❣ q✉→t ❝õ❛ ♠ët t➟♣ ♠ð✱ t❤➟♠ ❝❤➼ tr➯♥ ♠ët ✤÷í♥❣ t❤➥♥❣✳ ❚❤❛② ✈➔♦ ✤â✱ ỉ♥❣ ❝❤➾ ♥â✐ ✤➳♥ ♠ët ✤✐➸♠ ✧❜➯♥ tr♦♥❣✧ ♠ët ❦❤♦↔♥❣✭ ❈❛♥t♦r✱ ✶✽✼✷✱ tr ✾✽✮ ❤♦➦❝ ✧ ♥❤ú♥❣ ✤✐➸♠ tr♦♥❣✧ ❝õ❛ ♠ët t➟♣ ✤✐➸♠ ❧✐➯♥ tö❝✭❈❛♥t♦r✱ ✶✽✼✾✱ tr✳✶✸✺✮✳ ❚✉② ♥❤✐➯♥ ✤à♥❤ ♥❣❤➽❛ ❝õ❛ ✤✐➸♠ tr♦♥❣ ❝õ❛ tr ợ P ữ r❛ tr♦♥❣ t→❝ ♣❤➞♠ ✧●❡♦♠❡tr✐❝ ❆♣♣❧✐❝❛t✐♦♥s ♦❢ t❤❡ ■♥❢✐♥✐t❡s✐♠❛❧ ❈❛❧❝✉❧✉s✧✭♣❡❛♥♦✱ ✶✽✽✼✮ P❡❛♥♦ ①❡♠ ①➨t ♠ët t➟♣ ✤✐➸♠ A✭ ❚r♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ✶✱ ✷✱ ❤♦➦❝ ✸ ❝❤✐➲✉✮ ✈➔ ✤à♥❤ ♥❣❤➽❛ ♠ët ✤✐➸♠ ♣ ❧➔ ✧✤✐➸♠ tr♦♥❣✧ ♥➳✉ ❝â ♠ët sè ❞÷ì♥❣ r s❛♦ ❝❤♦ t➜t ❝↔ ❝→❝ ✤✐➸♠ ❝â ❦❤♦↔♥❣ ❝→❝ tø ♣ ✤➳♥ ❝❤ó♥❣ ♥❤ä ❤ì♥ r t❤➻ t❤✉ë❝ A✳ ✹✳✶✳✸✳ ❇♦r❡❧✱ ❇❛✐r❡ ✈➔ sü r❛ ✤í✐ ❝õ❛ t➟♣ ♠ð✳ ỵ t tr t ổ t t rớ ợ t rs ỵ ♥➔② sû ❞ư♥❣ ❦❤ët ❦❤→✐ ♥✐➺♠ ✤÷đ❝ ❣å✐ ❧➔ ✧❝♦♠♣❛❝t♥❡ss✧✿ ▼ët t➟♣ ❊ ✤÷đ❝ ❣å✐ ❧➔ ❝♦♠♣❛❝t ♥➳✉✱ ❝❤♦ ❤å ❜➜t ❦ý S ❝õ❛ t➟♣ ♠ð ❝❤ù❛ E ✱ ❝â ♠ët sè ❤ú✉ ❤↕♥ t➟♣ ❝♦♥ ❝õ❛ S ❝❤ù❛ E ỵ s õ t r t➟♣ ✤â♥❣ ✈➔ ❜à ❝❤➦♥ ❝õ❛ sè t❤ü❝ ❧➔ ❝♦♠♣❛❝t✳ ❑❤→✐ ♥✐➺♠ ❝õ❛ ♠ët t➟♣ ♠ð ❧➔ ❝➛♥ t❤✐➳t ✤➸ ♣❤→t ❜✐➸✉ t➼♥❤ ❝♦♠♣❛❝✳ ✈➔ ❞♦ ✤â ❝➛♥ t❤✐➳t ❝❤♦ ỵ tr qt ♥❤✐➯♥✱ ❦❤✐ ❇♦r❡❧ ♣❤→t ❜✐➸✉ ♣❤✐➯♥ ❜↔♥ ✤➛✉ t✐➯♥ ❝õ❛ ✤à♥❤ ❧➼ tr♦♥❣ ❧✉➟♥ →♥ t✐➳♥ ❞➽ ❝õ❛ ♠➻♥❤ æ♥❣ ✤➣ ❦❤æ♥❣ ✤➲ ❝➟♣ ✤➳♥ ❝→❝ t➟♣ ♠ð✳ ❑❤→✐ ♥✐➺♠ ✈➲ ♠ët t➟♣ ♠ð ✤➣ ✤÷đ❝ ♥➯✉ ❧➯♥ ❧➛♥ ✤➛✉ t✐➯♥ tr♦♥❣ ♠ët ❜↔♥ ✐♥✱ ❜è♥ ♥➠♠ s❛✉ ❦❤✐ ❇♦r❡❧ ự ỵ ổ ✤➛✉ t✐➯♥ ✤÷đ❝ ❇❛✐r❡ ①✉➜t ❜↔♥ tr♦♥❣ ❧✉➟♥ →♥ t✐➳♥ s➽ ❝õ❛ ♠➻♥❤ ❦❤✐ t❤↔♦ ❧✉➟♥ ✈➲ ❝→❝ ❤➔♠ t❤ü❝ ♥û❛ ❧✐➯♥ tö❝✳ ❙❛✉ ❦❤✐ ✤à♥❤ ♥❣❤➽❛ ♠ët ✧q✉↔ ❝➛✉ ✤â♥❣✧ S ✈➔✧ q✉↔ ❝➛✉ ♠ð✧ S ❝ò♥❣ t➙♠ ✈➔ ❜→♥ ❦➼♥❤ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❊❯❝❧✐❞❡ ♥ ❝❤✐➲✉✱ æ♥❣ ✈✐➳t✿ ❱ỵ✐ ✤✐➸♠ ❜➜t ❦ý ❝õ❛ S ✱ ❝â ♠ët q✉↔ ❝➛✉ ❜→♥ ❦➼♥❤ ❞÷ì♥❣ ♥❤➟♥ ✤✐➸♠ ◆❣✉②➵♥ ❚❤à ◆❣å❝ ❆♥❤ ✶✷ ❈❛♦ ❤å❝ ❑❤â❛ ✷✽ ❚♦→♥ ✹✳ P❍❹◆ ❚➑❈❍ ❚❘■ ❚❍Ù❈ ▲❯❾◆ ❈Õ❆ ❑❍⑩■ ◆■➏▼ ❚❾P ▼Ð ✤â ❧➔♠ t➙♠ ♠➔ t➜t ❝↔ ❝→❝ ✤✐➸♠ ❝õ❛ ♥â ✤➲✉ t❤✉ë❝ S ✳ ❚ê♥❣ q✉→t ❤ì♥✱ tỉ✐ ❣å✐ ♠å✐ t➟♣ ❤đ♣ ❝→❝ ✤✐➸♠ ❝â t➼♥❤ ❝❤➜t ♥➔② ❧➔ ♠ët ♠✐➲♥ ♠ð ❝â ♥ ❝❤✐➲✉✳ ◆❤÷ ✈➟②✱ ❇❛✐r❡ ✤à♥❤ ♥❣❤➽❛ t➟♣ ♠ð q✉❛ ❝→❝ q✉↔ ❝➛✉ ♠ð tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❊✉❝❧✐❞❡ ♥ ❝❤✐➲✉ ự ỵ ỷ tử tr ❉♦ ✤â q✉❛♥ ♥✐➺♠ ❝õ❛ ❇❛✐r❡ ✈➲ t➟♣ ♠ð ❧➔ q✉❛♥ ♥✐➺♠ ❤➻♥❤ ❤å❝✱ ❝â ❤➻♥❤ t❤ù❝ ❦❤❛✐ ♥✐➺♠ t♦→♥ ❤å❝ ✈➔ ❝â ❝ì ❝❤➳ ❝ỉ♥❣ ❝ư t÷í♥❣ ♠✐♥❤✳ ❑➳t ❧✉➟♥✿ ❚➟♣ ♠ð✱ ♣❤➛♥ tr♦♥❣ ❧➔ ♥❤ú♥❣ ❦❤→✐ ♥✐➺♠ ❝ì sð q✉❛♥ trå♥❣ tr♦♥❣ ❝→❝ ❦❤ỉ♥❣ ❣✐❛♥ ♥❤÷ ❦❤ỉ♥❣ ❣✐❛♥ t♦♣♦✱ ❦❤ỉ♥❣ ❣✐✱ ♥â ù♥❣ ❞ư♥❣ t↕♦ t✐➲♥ ✤➲ ❝❤♦ ự ỵ t t tr ♠ỉ♥ ❣✐↔✐ t➼❝❤✳ ❱➻ t❤➳ ♥✐➺♠ ♥➢♠ rã ✤÷đ❝ ❝→❝ ❦✐➳♥ t❤ù❝ ✈➲ t➟♣ ♠ð ♣❤➛♥ tr♦♥❣ ❧➔ r➜t q✉❛♥ trå♥❣✳ ◆❣✉②➵♥ ❚❤à ◆❣å❝ ❆♥❤ ✶✸ ❈❛♦ ❤å❝ ❑❤â❛ ✷✽ ❚♦→♥ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❬✶❪ ▼ët ♣❤➙♥ t➼❝❤ tr✐ t❤ù ❧✉➟♥ ❦❤→✐ ♥✐➺♠ t➟♣ ♠ð✱ t➟♣ ✤â♥❣ tr♦♥❣ ❣✐↔✐ t➼❝❤ ✈➔ t♦♣♦ ❤å❝✱ ◆❣✉②➵♥ ⑩✐ ◗❯è❝✱ ❱ã ❚❤à ❚ó ◗✉ý♥❤ ❬✷❪ ❇➔✐ ❣✐↔♥❣ ❚♦♣♦ ❈❛♦ ❤å❝ ✷✵✷✵✱ ❬✸❪ ●✐→♦ tr➻♥❤ t♦♣♦ ✤↕✐ ❝÷ì♥❣ ✶✹