Báo cáo nghiên cứu khoa học: "Không gian với sn-lưới sao-đếm được và sn-lưới sao-điểm" potx

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