A ‚" Rk
A " Sk
"µ>0"F "F"G"G(F ‚" A ‚" G (G \ F ) < µ)
"H"F "K"G (H ‚" A ‚" K '" (K \ H) = 0)
à ´
(i) Ò! (ii)
µ > 0
" "µ>0"F "F(F ‚" A '" |A \ F | < µ)
" "µ>0"G"G(A ‚" G '" |G \ A| < µ)
(G \ F ) = (G \ A) *" (A \ F )
 ‚"  x " (G \ F ) x " A x " (G \ A)
/
x " A x " (A \ F ) x " (G \ A) *" (A \ F )
 ƒ"  x " (G \ A) *" (A \ F ) x " (G \ A) x " F
/
F ‚" A x " (G \ F ) x " (A \ F )
x " G A ‚" G x " (G \ F )
F ‚" A ‚" G (G \ F ) d"
µ µ
(A \ F ) + (G \ A) < + = µ
2 2
(ii) Ò! (iii)
n " N Fn " F Gn " G
1
Fn ‚" A ‚" Gn (Gn \ Fn) <
n
G = Gn F = Fn F " FÃ G " G´
n"N n"N
F ‚" A "n(Fn ‚" A) A ‚" G "n(A ‚" Gn)
n " N (G \ F ) ‚" (Gn \ Fn) x " (G \ F )
x " Gn G ‚" Gn x " Fn Fn ‚" F x " (Gn \Fn)
/
1
n " N (G \ F ) d" (Gn \ Fn) <
n
n (G \ F ) = 0
(iii) Ò! (i)
A = H *" (A \ H) (A \ H) ‚" (K \ H)
A ‚" K
(K \ H) = 0
A \ H (A \ H) = 0 A
A " Sk
(A) = inf{(G) : A ‚" G '" G - otwarty}
(A) = sup{(F ) : F ‚" A '" F - domkniety}
(A) = sup{(F ) : F ‚" A '" F - zwarty}
A
A = An An = {x " A : x < n}
n"N
(A) = lim (An) An
n"
 Sk = Ã(B(Rk) *" Nk)
 ƒ" 
Nk ‚" Sk
B(Rk) ‚" Sk Sk Ã
B(Rk) *" Nk ‚" Sk Ã(B(Rk) *" Nk) ‚" Sk Sk
Ã
 ‚" 
A " Sk H " FÃ H ‚" A |A \ H| = 0
H ‚" A (A \ H) " Nk
" H " Nk *" B(Rk) H " FÃ ‚" B(Rk)
" (A \ H) " Nk *" B(Rk) (A \ H) " Nk
A = H *" (A \ H) " Ã(B(Rk) *" Nk) A
Sk ‚" Ã(B(Rk) *" Nk)
V ‚" [0, 1]
[0, 1] \ V
V *" [1, 2]
V *" A, gdzieA ‚" [0, 1]
V *" A, gdzieA ‚" [1, 2]
 [0, 1] \ V " S1 V =
[0, 1] \ ([0, 1] \ V ) " S1
V *" [1, 2] " S1
1 " V V = (V *" [1, 2]) \ [1, 2] " S1
/
1 " V V = (V *" [1, 2]) \ (1, 2] " S1
A = [0, 1] V *" A = [0, 1] " S1
A = V V *" A = V " S1
/
V *" A, gdzieA ‚" [1, 2] " S1
1 " V V = (V *" A) \ [1, 2] " S1
/
1 " V V = (V *" A) \ (1, 2] " S1