σ HCP 1 σ T 1 1.1 P X P a ∈ X, P a = {P ∈ P : a ∈ P } P a ∈ X, U a P  = {P ∈ P : P ∩ U = ∅} P P o ∈ P P(P o ) = {P ∈ P : P ∩P o = ∅} P = {P α : α ∈ Λ} cl(∪{B α : α ∈ Λ  }) = ∪{clB α : α ∈ Λ  }, Λ  ⊂ Λ B α ⊂ P α α ∈ Λ  clB B P {x(P ) ∈ P : P ∈ P} P σ P =  {P n : n ∈ N} P n n ∈ N. 1.2 V ⊂ X x ∈ V V x ∈ X {x n } X x n o ∈ N {x n : n ≥ n o } ⊂ V. 1 1.3 P =  {P x : x ∈ X} X P X P x x x ∈ X x ∈ ∩P x U x P ∈ P x P ⊂ U ∩P x ∩{P : P ∈ P x }; P 1 , P 2 ∈ P x P 3 ∈ P x P 3 ⊂ P 1 ∩ P 2 P x x P x x X X P =  {P x : x ∈ X} P x X X σ {P n } X {P n } X x ∈ X, {st(x, P n ) : n ∈ N} x X st(x, P n ) = ∪{P ∈ P n : x ∈ P }. 1.4 X P X P X K V K F P K ⊂ ∪F ⊂ V ∪F = ∪{P : P ∈ F}. P X {x n } X x ∈ X V x P ∈ P m ∈ N {x n : n  m} ∪ {x} ⊂ P ⊂ V. P cs ∗ X {x n } X x ∈ X V x {x n k } {x n } P ∈ P {x n k : k ∈ N} ∪ {x} ⊂ P ⊂ V. 1.5 X A X x ∈ A {x n } A {x n } x 1.6 X P X F P Int s (∪F) = {x ∈ X : ∪F x }. P x ∈ X U x F P x ∈ Int s (∪F) ⊂ ∪F ⊂ U; x ∈ ∩F. 1.7 P X P K ⊂ X {K α : α ∈ J} K {P α : α ∈ J} ⊂ P K = ∪{K α : α ∈ J} K α ⊂ P α α ∈ J. P X K X P ∗ ⊂ P P ∗ K X 1.8 X P ⊂ X {x n } P x n −→ x m ∈ N {x n : n  m} ∪ {x} ⊂ P. {x n } P {x n } P 2 2.1 X X X X (1) =⇒ (3) P =  {P x : x ∈ X} X P x x P ∈ P x x U X P ∈ P x x ∈ P ⊂ U x ∈ Int s (P ) x ∈ Int s (P ) ⊂ P ⊂ U. P P P x x ∈ X X (3) =⇒ (2) X P x ∈ X P x = {P ∈ P : x ∈ P }; (P x ) ∗ = {∪L : L P x }; G x = {G ∈ (P x ) ∗ : x ∈ Int s (G)} G =  {G x : x ∈ X}. P P x (P x ) ∗ x ∈ X {x n } X {x n } x ∈ X U X x ∈ U P L P x ∈ Int s (∪L) ⊂ ∪L ⊂ U x ∈ ∩L. G = ∪L G ∈ G x x n −→ x m ∈ N {x n : n  m} ⊂ ∪L. G ∈ G {x n : n  m} ∪ {x} ⊂ G ⊂ U. G X G G ∈ G x ∈ X P 1 , P 2 , . . . , P n ∈ P x G =  in P i G  ∈ G G  =  im P  i P  1 , P  2 , . . . , P  m ∈ P y , y X G  ∩P i = ∅ P  j P i ∩P  j = ∅ P P i P  ∈ P P  P P i G  ∈ G G G  ∈ G G (2) =⇒ (1) X X B = ∪{B x : x ∈ X} B x P B x B x = {B(x, n) : n ∈ N}, B(x, n + 1) ⊂ B(x, n) n P ∗ = {∩L : L P}. P ∗ P P P ∗ x ∈ X P P x = {P ∈ P : x ∈ P } P x = {P 1 , P 2 , . . . , P n , . . .} L x = {P ∈ P x : B(x, n) ⊂ P n }. L x = ∅ L x = ∅ m, n ∈ N B(x, n)  P m . {x n,m } x n,m ∈ B(x, n)\P m m, n ∈ N {x k } {x n,m } k n n k m x 1 = x 1,1 , x 2 = x 2,1 , x 3 = x 2,2 , x 4 = x 3,1 x 5 = x 3,2 , x 6 = x 3,3 , x 7 = x 4,1 , . . . . k = m + n(n − 1)/2 U x {B(x, n) : n ∈ N} x n ∈ N B(x, n) ⊂ U B(x, n + 1) ⊂ B(x, n) n n ∈ N k o ∈ N x k ∈ B(x, n) k  k o , n ∈ N x k −→ x P P m o ∈ P x k 1 ∈ N {x k : k  k 1 } ⊂ P m o . {x k } k k > k 1 x k = x n,m o n > k 1 x k /∈ P m o k > k 1 L x = ∅ x ∈ X L = ∪{L x : x ∈ X}. U X x ∈ U P L x = ∅ P ∈ P x n ∈ N B(x, n) ⊂ P ⊂ U L x x x ∈ ∩L x P P  L x B(x, n) ⊂ P B(x, n  ) ⊂ P  P P ∩ P  ∈ P n  = max{n, n  } B(x, n  ) ⊂ P ∩ P  P ∩ P  ∈ L x B B(x, n) x P ∈ L x x L X L ⊂ P P L 2.2 P = {P α : α ∈ Λ} X P ∗ P 2.3 X X cs ∗ σ HCP 2.4 X X σ HCP σ W HCP X σ HCP σ W HCP B = ∪{B x : x ∈ X} X B x P = ∪{P n : n ∈ N ∗ } P n n ∈ N ∗ B x = {B x,1 , B x,2 , . . . , B x,n , . . .} B x,n+1 ⊂ B x,n n P n ⊂ P n+1 n (P n ) ∗ (P n ) ∗ P n P n x ∈ X {P ∈ P x : ∃B(x, n) B(x, n) ⊂ P } = ∅, P x = {P ∈ P : x ∈ P } P n ⊂ P n+1 n L n,x = {P ∈ P n : ∃B ∈ B x , B ⊂ P } = ∅, n L n,x = ∅ n L x = ∪{L n,x : n = 1, 2, . . .}, L n = ∪{L n,x : x ∈ X}, L = ∪{L x : x ∈ X}. L n ⊂ P n n L = ∪{L n : n = 1, 2, . . .} ⊂ P. P P n L σ − HCP L L x L x x x ∈ X P P  L x B x,n B x,n  B x B x,n ⊂ P, B x,n  ⊂ P  m = max{n, n  } B x,m ⊂ B x,n ∩ B x,n  ⊂ P ∩ P  ∈ P m . P ∩ P  ∈ L m,x ⊂ L x . B x,n x P ∈ L x x L X P σ W HCP 2.5 X x ∈ X x U x ∈ X U x x ∈ X \ U X {x n } ⊂ X \ U x n −→ x U x n o ∈ N {x n : n  n o } ⊂ U. U x 2.6 P cs ∗ σ HCP X P 2.7 2.8 P X P P X P o P P o P ∈ P o A P ⊂ P A = {A P : P ∈ P o }. P o A cl(∪A) = ∪{clA : A ∈ A}. cl(∪A) ⊂ ∪{clA : A ∈ A}. x ∈ cl(∪A) A U x A x = {A ∈ A : A ∩ U = ∅} U ∩ (∪(A \ A x )) = ∅ . U U ∩ cl(∪(A \ A x )) = ∅ . x ∈ cl(∪A) = cl(∪(A \ A x )) ∪ cl (∪A x ) x ∈ cl(∪A x ) = ∪{clA : A ∈ A x }. cl(∪A) ⊂ ∪{clA : A ∈ A}. P 2.9 P L ∪P P ∈ P L P L P 2.10 X X X {G n } G n+1 G n n ∈ N G n n ∈ N (1) =⇒ (2) X X J = ∪{J n : n ∈ N} J n X J = ∪{J x : x ∈ X} J x x n = 1, 2, . . . K n = {x ∈ X : J x ∩ J n = ∅}, P n = J n ∪ {K n } G n = {G = ∩{P i : i  n, P i ∈ P i }}. {G n } G n X G n+1 G n n ∈ N x ∈ X U X x ∪{J t : t ∈ X} P ∈ J x P ⊂ U ∪{J n : n ∈ N} = ∪{J t : t ∈ X} n ∈ N P ∈ J n ⊂ P n . J n J n ∩ J x = {P} x /∈ K n st(x, G n ) ⊂ P ⊂ U, st(x, G n ) = ∪{G ∈ G n : x ∈ G} {st(x, G n ) : n ∈ N} x G n+1 G n n ∈ N st(x, G l ) ⊂ st(x, G n ) ∩ st(x, G m ) l > max{n, m} {st(x, G n ) : n ∈ N} S X x ∈ st(x, G n ) J x ∩ J n = ∅ P ∈ J x ∩ J n P x S P S st(x, G n ) J x ∩ J n = ∅ U = X \ ∪{P ∈ J n : x /∈ P }. x ∈ U J n V 1 x V 1 J n V x V ⊂ U U x U ⊂ st(x, G n ) S st(x, G n ) st(x, G n ) x {st(x, G n ) : n ∈ N} x {G n } J n P n G n n {G n } G n C X x ∈ C V x x V x J n C C F 1 , F 2 , . . . , F k J n C j = F j ∩ C, j = 1, 2, . . . , k K = C \ (∪{int C C j : j = 1, 2, . . . , k}), int C C j C j C X X σ B B σ HCP B C C C x ∈ C F ∈ J x F x x ∈ int C (F ∩ C) K ⊂ K n C = ( k  j=1 C j ) ∪ K, C j ⊂ F j , j = 1, 2, . . . , k K ⊂ K n F j C j K P n G n (1) =⇒ (2) X {G n } U = ∪{U x : x ∈ X} X U x = {st(x, G n ) : n ∈ N} U x X X X cs ∗ σ HCP G = ∪{G n : n ∈ N} σ G σ HCP {x n } X x ∈ X U x {st(x, G n ) : n ∈ N } x n ∈ N x ∈ st(x, G n ) ⊂ U. st(x, G n ) x {x n } st(x, G n ) G ⊂ G n x ∈ G {x n } G {x n k } {x n } {x n k } ⊂ G G cs ∗ X X cs ∗ σ HCP σ HCP