(X,τ ) F ⊆ X ˚ F = ∅ F F ⊆ X F ⊆ X X ∀A ⊆ X A = ∅ A X ⇐⇒∀{Un} Un X Un = ∅ X {Un} Un = ∅ Un = X Un X Un = Un Un = X ˚ A = A ˚ Un = ˚ Un = Un = Un = X = ∅ X X {Fn} X = Fn Fn Fn = ∅ Fn
X X d : X × X → [0, ∞)
• ∀x ∈ X d(x,x)=0
• ∀x,y ∈ X d(x,y)= d(y,x)
• ∀x,y,z ∈ X d(x,z) ≤ d(x,y)+ d(y,z) (X,d) (X,d) x ∈ X r ∈ [0, ∞) r x B(x,r)= {y ∈ X|d(x,y) <r}
X ∀{
X Un (1) ⇒ (2) X {Un} Un A ⊆ X A ∩ Un = ∅ {Un ∩ A} A (Un ∩ A)= ∅ A (2) ⇒ (1) A ⊆ X {Un} A Un = ∅ B = X\A {Un ∪ B} X V ⊆ X V V ⊆ A V A W W A Un A {Un ∪ B} X (Un ∪ B
Un
= B B X A
Un}
)=(
∪ B)= ∅∪ B
d (X,d) {xn}⊆ X ∀ > 0 ∃N ∈ N ∀n,m ≥ N d(xn,xm) < X (X,d) (X,d) {Fn}n>0 X = Fn ˚ Fn = ∅ F1 = X F1 = ∅ x1 ∈ F1 r1 ∈ R 0 <r1 < 1 2 B (x1,r1) ⊂ F1 ˚ F2 = ∅ B x1, r1 2 ⊂ F2 B x1, r1 2 ∩ F2 = ∅ x2 ∈ B x1, r1 2 ∩ F2 r2 ∈ R 0 <r2 < 1 22 B (x2,r2) ⊂ B x1, r1 2 ∩ F2 {B (xn,rn)}n>0 • B (xn+1,rn+1) ⊂ B xn, rn 2 • B (xn,rn) ∩ Fn = ∅
• 0 <rn < 1 2n n<m d(xn,xm) ≤ d(xn,xn+1)+ d(xn+1,xn+2)+ + d(xm 1,xm) ≤ ≤ rn 2 + rn+1 2 + ... + rm 1 2 ≤ ≤ 1 2n+1 + 1 2n+2 + ... + 1 2m = = 1 2n+1 1+ 1 2 + ... + 1 2m n 1 < < 1 2n {xn} X x ∈ X n m>n d(x,xn) ≤ d(x,xm)+ d(xm,xn) <d(x,xm)+ rn 2 m →∞ d(x,xn) <rn x ∈ B (xn,rn) B (xn,rn) ∩ Fn = ∅ x Fn x ∈ X = Fn X {Un} X Un A ⊆ X A ∩ Un = ∅ {A ∩ Un} A A\ (A ∩ Un) A A X A = A\∅ = A\ (A ∩ Un)= A\ (A ∩ Un) (X,τ ) ⇒ X
R R {x} ˚ {x} = ∅ R Y X = Y Q Y Y Q R Y p ∈ Y X = Y ∪{p} τ B = {{x} : x ∈ Y }∪{X\F : F ⊂ Y,card (F ) < ∞} (X,τ ) Y X X
R\Q (X,τ ) X X X