Báo cáo nghiên cứu khoa học: "Về không gian S-đóng đếm được" doc

S
S s S
s
T
0
T
1
T
2
S
s
S
s
S s S
s
(X, τ) A X A
U U ⊂ A ⊂ cl(U )
(X, τ)
SO(X , τ) SC(X, τ)
A
A scl(A) A
A sint(A) scl(A)
A sint(A) A
A (X, τ)
A
SR(X, τ) (X, τ)
1
(X, τ)
A (X, τ )
sint(A) = A
A ⊂ cl(int(A))

(X, τ)
A (X, τ)
scl(A) = A
int(cl(A)) ⊂ A
(X, τ)
scl(A) ∈ SR(X, τ) A ∈ SO(X, τ)
int(cl(A)) = scl(A) A ∈ τ
A (X, τ)
A = cl(int(A))
RC(X, τ ) (X, τ)
A (X, τ) A
A
1
, A
2
∈ RC(X, τ ) A
1
∪ A
2
∈ RC(X, τ )
A
1
, A
2
∈ RC(X, τ ) A
1
= cl(int(A
1
)) A
2

= cl(int(A
2
))
A
1
∪ A
2
= cl(int(A
1
) ∪ int(A
2
)) ⊂ cl(int(A
1
∪ A
2
)) A
1
∪ A
2
int(A
1
∪ A
2
) ⊂ A
1
∪ A
2
cl(int(A
1
∪ A

2
)) ⊂ A
1
∪ A
2
A
1
∪ A
2
=
cl(int(A
1
∪ A
2
)) A
1
∪ A
2
∈ RC(X, τ ). 
A (X, τ)
X\A
RO(X, τ) (X, τ)
A (X, τ) A
A (X, τ) A = int(cl(A))
A
1
, A
2
∈ RO(X, τ) A
1

∩ A
2
∈ RO(X, τ)
A
1
, A
2
∈ RO(X, τ) A
1
= int(cl(A
1
)) A
2
= int(cl(A
2
))
int(cl(A
1
∩A
2
)) ⊂ int(cl(A
1
)∩cl(A
2
)) = A
1
∩A
2
. A
1

∩A
2
A
1
∩A
2
⊂
cl(A
1
∩ A
2
) A
1
∩ A
2
⊂ int(cl(A
1
∩ A
2
)) A
1
∩ A
2
= int(cl(A
1
∩ A
2
))
A
1

∩ A
2
∈ RO(X, τ). 
(X, τ)
A ∈ RO(X, τ) A ∈ SO(X, τ) cl(A) ∈ R C(X, τ)
A ∈ RC(X, τ ) A ∈ SC(X, τ ) int(A) ∈ RO(X, τ)
A ∈ RO(X, τ) A = int(cl(A)). cl(A) =
cl(int(cl(A))) cl(A) ∈ RC(X, τ)
A ∈ SO(X, τ) A ⊂ cl(int(A)) cl(A) ⊂
cl(int(A)) ⊂ cl(int(cl(A))) int(cl(A)) ⊂ cl(A) cl(int(cl(A))) ⊂ cl(A)
cl(A) = cl(int(cl(A))) cl(A) ∈ RC(X, τ)
A ∈ RC(X, τ ) A = cl(int(A)) int(A) = int(cl(int(A)))
int(A) ∈ RO(X, τ)
A ∈ SC(X, τ ) int(cl(A)) ⊂ A int(cl(int(A))) ⊂
int(cl(A)) ⊂ int(A) int(A) ⊂ cl(int(A)) int(A) ⊂ int(cl(int(A)))
int(A) = int(cl(int(A))) int(A) ∈ RO(X, τ). 
A (X, τ)
U U ⊂ A ⊂ cl(U )
(X, τ)
(X, τ)
(X, τ)
F (X, τ)
U F = cl(U)
U F = cl(U)
U F = cl(U)
U F = cl(U)
⇒ F
(X, τ) F = cl(int(F )) U = int(F ) U
F = cl(U )
⇒

⇒
⇒ U F = cl(U)
F
⇒ F (X, τ ) int(F ) = U
U F = cl(U )
⇒ U F = cl(U) U
V V ⊂ U ⊂ cl(V ) F = cl(V )
F 
(X, τ)
A (X, τ )
F A = int(F )
F A = int(F )
F A = int(F )

A (X, τ )
A ⊂ int(cl(A))
(X, τ) A
(X, τ) A
S s
(X, τ) S S
{U
α
: α ∈ ∧} (X, τ ) ∧
0
∧ X = ∪{cl(U
α
) : α ∈ ∧
0
}
(X, τ)

(X, τ) S
(X, τ)
U = {V
α
: α ∈ ∧} (X, τ)
∧
0
∧ X = ∪{cl(V
α
) : α ∈ ∧
0
}
⇒ (X, τ) S {U
α
: α ∈ ∧}
(X, τ)
{U
α
: α ∈ ∧} (X, τ) (X, τ) S
∧
0
∧ X = ∪{cl(U
α
) : α ∈ ∧
0
}
cl(U
α
) = U
α

α ∈ ∧
0
X = ∪{U
α
: α ∈ ∧
0
}
⇒ {V
α
: α ∈ ∧} (X, τ)
{cl(V
α
) : α ∈ ∧} (X, τ)
∧
0
∧ X = ∪{cl(V
α
) : α ∈ ∧
0
}
⇒ {U
α
: α ∈ ∧} (X, τ)
{cl(U
α
) : α ∈ ∧} (X, τ)
∧
0
∧ X = ∪{cl (U
α

) : α ∈ ∧
0
}
(X, τ) S 
(X, τ) A
X RC(A, τ
A
) = {F ∩ A : F ∈ RC(X, τ )} τ
A
τ A
(X, τ) B S
X (X, τ) S
{F
α
: α ∈ ∧} X
{B ∩ F
α
: α ∈ ∧} B
B B S ∧
0
∧
B = ∪{B ∩ F
α
: α ∈ ∧
0
} X = cl(B) = ∪{cl(B ∩ F
α
) : α ∈ ∧
0
} ⊂ ∪{cl (F

α
) :
α ∈ ∧
0
} = ∪{F
α
: α ∈ ∧
0
}. X = ∪{F
α
: α ∈ ∧
0
} (X, τ) S 
(X, τ)
U ∈ τ cl(U ) ∈ τ
(X, τ)
(X, τ)
(X, τ)
(X, τ)
RO(X, τ ) = RC(X, τ)
cl(U ) = scl(U ) U (X, τ)
(X, τ)
(X, τ) S
(X, τ)
{U
α
: α ∈ ∧} (X, τ )
{U
α
: α ∈ ∧} (X, τ) (X, τ)

∧
0
∧ X = ∪{U
α
: α ∈ ∧
0
} (X, τ) S 
(X, τ) H H
{U
α
: α ∈ ∧} X ∧
0
∧ X = ∪{cl(U
α
) : α ∈ ∧
0
}
(X, τ) (X, τ ) H
(X, τ) s s
{U
α
: α ∈ ∧} (X, τ ) ∧
0
∧ X = ∪{scl(U
α
) : α ∈ ∧
0
}
(X, τ) s (X, τ) S
(X, τ) s

X
f : (X, τ) −→ (Y, σ) (X, τ )
(Y, σ) f
−1
(U) ∈ SO(X, τ )
U ∈ SO(Y, σ)
f : (X, τ) −→ (Y, σ) (X, τ)
(Y, σ)
f
scl(f
−1
(B)) ⊂ f
−1
(scl(B)) B ⊂ Y
(X, τ) s f : (X, τ) −→ (Y, σ)
(Y, σ) s
(X, τ) s f : (X, τ) −→ (Y, σ)
{U
α
: α ∈ ∧} Y
Y f {f
−1
(U
α
) : α ∈ ∧} X
X {scl(f
−1
(U
α
)) : α ∈ ∧} X

(X, τ) s ∧
0
∧
X =

α∈∧
0
scl(f
−1
(U
α
)) f
scl(f
−1
(U
α
)) ⊂ f
−1
(scl(U
α
)) α ∈ ∧
0
Y = f(X) =

α∈∧
0
f(scl(f
−1
(U
α

))) ⊂

α∈∧
0
f(f
−1
(scl(U
α
))) =

α∈∧
0
scl(U
α
) =

α∈∧
0
U
α
.
Y =

α∈∧
0
U
α
(Y, σ) s 
f : (X, τ) −→ (Y, σ) (X, τ )
(Y, σ) ν ν f

−1
(U)
(X, τ) U (Y, σ).
(X, τ) f : (X, τ) −→ (Y, σ)
ν (Y, σ) s
(X, τ) f : ( X, τ ) −→ (Y, σ)
ν {U
α
: α ∈ ∧} Y
{f
−1
(U
α
) : α ∈ ∧} X (X, τ)
∧
0
∧ X =

α∈∧
0
f
−1
(U
α
) f
Y = f(X) = f


α∈∧
0

f
−1
(U
α
)

=

α∈∧
0
f(f
−1
(U
α
)) =

α∈∧
0
U
α
.
(Y, σ) s 
s S
(X, τ) f : (X, τ) −→ (Y, σ) ν
(Y, σ) S
S
(X, τ) S
S X
S S
(X, τ) S

{U
n
: n = 1, 2, . . . ) X I
{1, 2, . . . } X =

n∈I
cl(U
n
)
(X, τ)
{U
n
: n = 1, 2, . . . } X
I {1, 2, . . . } X = ∪{cl(U
n
) : n ∈ I}
(X, τ) S (X, τ)
(X, τ) P P
G
δ
(X, τ)
(X, τ) P S
X τ = {∅} ∪ {A ⊂ X : X\A
} τ X (X, τ)
P S τ
X ∅ X τ A
1
∈ τ A
2
∈ τ A

1
= ∅ A
2
= ∅
A
1
∩ A
2
= ∅ ∈ τ A
1
= ∅ A
2
= ∅ X\A
1
X\A
2
X\(A
1
∩ A
2
) = (X\A
1
) ∪ (X\A
2
) A
1
∩ A
2
∈ τ
{A

i
: i ∈ I} τ A
i
= ∅ i ∈ I
∪{A
i
: i ∈ I} = ∅ ∈ τ i
0
∈ I A
i
0
= ∅ X\A
i
0
X\ ∪ {A
i
: i ∈ I} = ∩{X\A
i
: i ∈ I} ⊂ X\A
i
0
X\ ∪ {A
i
: i ∈ I}
∪{A
i
: i ∈ I} ∈ τ τ X
(X, τ) P G G
δ
X G = ∩{U

n
: n = 1, 2, . . . } U
n
∈ τ n n
0
U
n
0
= ∅ G = ∅ ∈ τ U
n
= ∅ n X\U
n
n U
n
= X\B
n
B
n
∩{U
n
: n = 1, 2, . . . } =
∩{X\B
n
: n = 1, 2, . . . } = X\ ∪ {B
n
: n = 1, 2, . . . } B
n
n
∪{B
n

: n = 1, 2, . . . } X\ ∩ {U
n
: n = 1, 2, . . . }
G = ∩{U
n
: n = 1, 2, . . . } ∈ τ (X, τ) P
(X, τ) S F
(X, τ) A ∈ τ F = cl(A) A = ∅ F = ∅
A = ∅ X\A A = X\B B
cl(A) = cl(X\B) = X\int(B) int(B) = ∅ int(B) ∈ τ
X\int(B) X int(B)
X\int(B) int(B) = ∅
F = cl(A) = X F X F = ∅
F = X X
(X, τ) S 
P (X, τ ) S
(X, τ)
(X, τ) S
X
(X, τ) S
X τ = {∅} ∪ {A ⊂ X : X\A
} τ
X (X, τ) S (X, τ)
X
X = {x
1
, x
2
, . . . , x
n

, . . . } n = 1, 2, . . . A
n
= {x
1
, x
2
, . . . , x
n
}
U
n
= {X\A : A ∈ P(A
n
)} P(A
n
) = {A : A ⊂ A
n
} U
n
n U =
∞

n=1
U
n
U τ U
(X, τ)
(X, τ)
(X, τ) x, y ∈ X x = y
U x V y U ∩ V = ∅ U ∈ τ V ∈ τ X\U X\V

(X\U ) ∪ (X\V ) = X\(U ∩ V ) = X
X (X, τ )

(X, τ)
(X, τ)
X
(X, τ)
(X, τ) S
(X, τ)
{F
n
: n = 1, 2, . . . } (X, τ)
(X, τ) {F
n
: n = 1, 2, . . . }
(X, τ) (X, τ )
I {1, 2, . . . } X =

n∈I
F
n
(X, τ) S 
f : (X, τ) −→ (Y, σ) (X, τ )
(Y, σ)
f
−1
(V ) ∈ τ V ∈ SO(Y, σ)
(X, τ) f : (X, τ) −→
(Y, σ) (Y, σ) S
(X, τ) f : (X , τ) −→ (Y, σ)

{U
n
: n = 1, 2, . . . } (Y, σ)
{f
−1
(U
n
) : n = 1, 2, . . . } (X , τ)
(X, τ) I {1, 2, . . . }
X =

n∈I
f
−1
(U
n
)
Y = f(X) = f


n∈I
f
−1
(U
n
)

=

n∈I

U
n
⊂

n∈I
cl(U
n
) ⊂ Y.
Y =

n∈I
cl(U
n
) (Y, σ) S 
(X, τ) s
s {U
n
: n = 1, 2, . . . } X
I {1, 2, . . . } X = ∪{scl(U
n
) : n ∈ I}
s s
s S
(X, τ) s
X
(X, τ) A
(X, τ) SR(A, τ
A
) = A ∩ SR(X, τ) τ
A

τ A
(X, τ) s U ∈ RO(X, τ)
U s (X, τ)
(X, τ) s U ∈ RO(X, τ) {A
n
:
n = 1, 2, . . . } U U U ∈ RO(X, τ)
U n ∈ {1, 2, . . . }
F
n
∈ SR(X, τ) A
n
= U ∩ F
n
X\U ∈ RC(X, τ ) X\U ∈ SR(X, τ)
{X\U } ∪ {F
n
: n = 1, 2, . . . } (X, τ)
I {1, 2, . . . }
X = (X\U ) ∪


n∈I
F
n

.
U ⊂

n∈I

F
n
U =

n∈I
A
n
U s
(X, τ). 
f : (X, τ) −→ (Y, σ)
(X, τ) (Y, σ)
f f(U) ∈ SO(Y, σ) U ∈
SO(X, τ)
f f(F ) ∈ SC(Y, σ)
F ∈ SC(X, τ )
f : (X, τ) −→ (Y, σ) f
f : (X, τ) −→ (Y, σ) (X, τ)
s (Y, σ) s
f : (X, τ ) −→ (Y, σ) (Y, σ) s
(X, τ) s
f : (X, τ) −→ (Y, σ) (X, τ)
s (Y, σ) s {U
n
: n =
1, 2, . . . } Y {f
−1
(U
n
) : n = 1, 2, . . . }
X (X, τ) s

I {1, 2, . . . } X =

n
∈
I
scl(f
−1
(U
n
)) f
scl(f
−1
(U
n
)) ⊂ f
−1
(scl(U
n
)) n = 1, 2, . . .
X ⊂

n∈I
f
−1
(scl(U
n
)) Y = f(X) ⊂

n∈I
scl(U

n
) Y =

n∈I
scl(U
n
)
Y s
f : (X, τ ) −→ (Y, σ) (Y, σ)
s (X, τ) s
{U
n
: n = 1, 2, . . . } X
f {f(U
n
) : n = 1, 2, . . . }
Y f f
f(U
n
) n = 1, 2, . . . {f(U
n
) : n = 1, 2, . . . }
Y (Y, σ) s
I {1, 2, . . . } Y =

n∈I
f(U
n
)
X = f

−1
(Y ) =

n∈I
U
n
X s 
(X, τ) s
s {U
n
: n = 1, 2, . . . } (X, τ )
I {1, 2, 3, . . . } X = ∪{scl(U
n
) : n ∈ I}
f : (X, τ) −→ (Y, σ) f
f : (X, τ) −→ (Y, σ)
(X, τ) s (Y, σ) s
f : (X, τ ) −→ (Y, σ) (X, τ) s
(Y, σ) s {U
n
: n = 1, 2 . . . }
Y f {f
−1
(U
n
) : n = 1, 2, . . . }
X X s I
{1, 2, . . . } X =

n∈I

scl(f
−1
(U
n
)) f
f scl(f
−1
(U
n
)) ⊂ f
−1
(scl(U
n
))
n = 1, 2, . . . Y = f(X) =

n∈I
scl(U
n
) (Y, σ) s 
[1]
[2] s
[3] S
[4]
[5] s
[6] S
[7]
[8]
[9] S
S

S s
S s