Research report: "the probability measure functor preserving some topological properties that" ppt

P
k
(X) X
P
k
(X)
1. Introduction and notions
cs cs
∗
X
X µ : X → [0, 1]
(1) µ = {x ∈ X : µ(x) > 0}
(2)

x∈ µ
µ(x) = 1
k ∈ N P
k
(X) X
k µ ∈ P
k
(X)
µ =
q

i=1
m
i
δ
x
i

q  k δ
x
(y) δ
x
(y) =

0 y = x
1 y = x,
m
i
= µ(x
i
) > 0,
q

i=1
m
i
= 1 m
i
= µ(x
i
) µ x
i
P
k
(X)
µ ∈ P
k
(X) µ =


q
i=1
m
i
δ
x
i
µ
O(µ, U
1
, U
2
, . . . , U
q
, ε) =
{µ ∈ P
k
(X) : µ =
q+1

i=1
µ
i
, µ
i
⊂ U
i
, | µ
i

 −m
i
|< ε, i = 1, 2, , q +1, m
q+1
= 0},
1
ε > 0, U
1
, U
2
, , U
q
x
1
, x
2
, . . . , x
q
U
q+1
= X\

q
i=1
U
i
 µ
i
=


x∈ µ
T
U
i
µ(x)
µ
i
i = 1, 2, , q + 1
 µ
i
= 1.
B X {O(µ, U
1
, , U
q
, ε) : µ ∈
P
k
(X); U
i
∈ B, i = 1, 2, , q; q  k, ε > 0} P
k
(X)
Fedorchuk topology
µ =
q

i=1
m
i

δ
x
i
∈ P
k
(X), µ
n
=
q

i=1
m
n
i
δ
x
n
i
, q  k
{µ
n
} µ
o
ε > 0 N > 0
n ≥ N x
n
i
∈ U
i
, i = 1, . . . , q |m

n
i
− m
o
i
| < ε, i = 1, . . . , q
X x ∈ P ⊂ X P
x {x
n
} x P
k ∈ N x
n
∈ P n ≥ k
X P X
(1) P X x ∈ X U
x P ∈ P x ∈ P ⊂ U.
(2) P cs X {x
n
} x ∈ X
U x m ∈ N P ∈ P
{x}

{x
n
: n ≥ m} ⊂ P ⊂ U
(3) P cs
∗
X {x
n
} ⊂ X

x ∈ X U x {x
n
i
: i ∈ N}
{x
n
} P ∈ P {x}

{x
n
i
: i ∈ N} ⊂ P ⊂ U
(4) P wcs
∗
X {x
n
}
x ∈ X U x {x
n
i
} {x
n
}
P ∈ P {x
n
i
: i ∈ N} ⊂ P ⊂ U
P =

x∈X

P
x
X P
x ∈ X
P
x
x
P
1
P
2
∈ P
x
P ∈ P
x
P ⊂ P
1
∩ P
2
(1) P X G ⊂ X G X x ∈ G
P ∈ P
x
P ⊂ G P
x
x
(2) P sn X P
x
x x ∈ X P
x
sn x

2. The main results
X
µ =
q

i=1
m
i
δ
x
i
∈ P
k
(X), q  k P
X P
i
∈ P, i = 1, 2, . . . q ε > 0 P
∗
µ
P
∗
µ
= P
∗
(µ, P
1
, P
2
, . . . , P
q

, ε) =
=

µ =
q+1

i=1
µ
i
: µ
i
⊂ P
i
, | µ
i
 −m
i
|< ε, i = 1, 2, . . . , q; P
q+1
= X \
q

i=1
P
i
,
m
q+1
= 0; P
i


P
j
= φ with i = j, and  µ
i
=

x
i
∈ µ
T
P
i
µ(x
i
),  µ
q+1
< ε

.
P
∗
µ
= {P
∗
(µ, P
1
, P
2
, . . . , P

q
, ε) : P
i
∈ P, i = 1, . . . , q, ε > 0}, P
∗
=

µ∈P
k
(X)
P
∗
µ
.
P
∗
P
k
(X)
P X P
∗
P
k
(X)
µ P
k
(X) µ =
q

i=1

m
i
δ
x
i
∈ P
k
(X), q  k O(µ, U
1
, U
2
, . . . , U
q
, ε)
µ P
∗
µ
∈ P
∗
µ ∈ P
∗
µ
⊂ O(µ, U
1
, . . . , U
q
, ε).
U
i
∩ U

j
= ∅ i = j P i = 1, . . . , q
P
i
∈ P x
i
∈ P
i
⊂ U
i
i = 1, . . . , q P
∗
µ
= P
∗
(µ, P
1
, . . . , P
q
, ε)
µ ∈ P
∗
µ
⊂ O(µ, U
1
, , U
q
, ε) P
∗
P

k
(X)
P =

x∈X
P
x
X P
(1) P
x
(2) P
1
, P
2
∈ P
x
P ∈ P
x
P ⊂ P
1

P
2
P
∗
=

µ∈P
k
(X)

P
∗
µ
P
∗
1
, P
∗
2
∈ P
∗
µ
P
∗
1
= P
∗
(µ, P
1
1
, . . . , P
1
q
, ε
1
) where ε
1
> 0 P
∗
2

=
P
∗
(µ, P
2
1
, . . . , P
2
q
, ε
2
), where ε
2
> 0 µ =
q

i=1
m
i
δ
x
i
∈ P
k
(X), q  k
P
1
i

P

1
j
= φ i = j P
1
i
∈ P
x
i
, i = 1, 2, . . . , q
P
2
i

P
2
j
= φ i = j P
2
i
∈ P
x
i
, i = 1, 2, . . . , q
P
1
i
∩ P
2
i
= ∅ i = 1, . . . , q P

P
3
i
∈ P
x
i
P
3
i
⊂ P
1
i

P
2
i
i = 1, . . . , q ε < min{ε
1
, ε
2
}
P
∗
3
= P
∗
(µ, P
3
1
, . . . , P

3
q
, ε) P
∗
3
⊂ P
∗
1

P
∗
2
.
P sn X P
∗
sn P
k
(X)
µ ∈ P
k
(X)
P
∗
= P
∗
(µ, P
1
, . . . , P
q
, ε) ∈ P

∗
µ
µ µ =
q

i=1
m
i
δ
x
i
m
i
= µ(x
i
) > 0 δ
x
i
i = 1, . . . , q {µ
n
}
µ P
k
(X) µ
n
=
q+1

i=1
µ

n
i
µ
n
−→ µ
ε > 0 n
i
∈ N µ
n
i
⊂ P
i
, | µ
n
i
 −m
i
|< ε, i =
1, 2, . . . , q;  µ
n
q+1
< ε n ≥ n
i
n
o
= max{n
1
, n
2
, . . . , n

q
}
n ≥ n
o
µ
n
∈ P
∗
(µ, P
1
, P
2
, . . . , P
q
, ε) P
∗
µ
P cs
∗
X P
∗
cs
∗
P
k
(X)
{µ
n
} µ P
k

(X) O(µ, U
1
, . . . , U
q
, ε)
µ µ =

q
i=1
m
i
δ
x
i
µ
n
=

q+1
i=1
µ
n
i
q  k
µ
n
−→ µ m ∈ N
µ
n
∈ O(µ, U

1
, . . . , U
q
, ε), for every n ≥ m.
n ≥ m µ
n
i
⊂ U
i
, | µ
n
i
 −m
i
|< ε, i = 1, 2, . . . , q
 µ
n
q+1
< ε i = 1, . . . , q n ≥ m x
n
i
∈ µ
n
i
µ
n
−→ µ {x
n
i
}

x
i
i = 1, . . . , q P cs
∗
X
x
n
i
−→ x
i
i = 1, . . . , q i = 1, . . . , q U
i
x
i
P
i
∈ P {x
n
j
i
} {x
n
i
} {x
n
j
i
}

{x

i
} ⊂ P
i
⊂ U
i
P
∗
= P
∗
(µ, P
1
, P
2
, . . . , P
q
, ε) n
j
µ
n
j
=

q+1
i=1
µ
n
j
i
µ
n

j
∈ P
k
(X)
{µ
n
j
}

{µ} ⊂ P
∗
= P
∗
(µ, P
1
, P
2
, . . . , P
q
, ε) ⊂ O(µ, U
1
, U
2
, . . . , U
q
, ε).
P cs X P
∗
cs P
k

(X)
{µ
n
} µ P
k
(X) O(µ, U
1
, . . . , U
q
, ε)
µ µ =

q
i=1
m
i
δ
x
i
µ
n
=

q+1
i=1
µ
n
i
q  k
µ

n
−→ µ m ∈ N
µ
n
∈ O(µ, U
1
, . . . , U
q
, ε), for every n ≥ m.
n ≥ m µ
n
i
⊂ U
i
, | µ
n
i
 −m
i
|< ε, i = 1, 2, . . . , q
 µ
n
q+1
< ε i = 1, . . . , q n ≥ m x
n
i
∈ µ
n
i
µ

n
−→ µ {x
n
i
}
x
i
i = 1, . . . , q
P cs X x
n
i
−→ x
i
i = 1, . . . , q
i = 1, . . . , q U
i
P
i
∈ P m
i
∈ N
{x
i
}

{x
n
i
: n ≥ m
i

} ⊂ P
i
⊂ U
i
.
m = max{m
1
, m
2
, . . . , m
q
} P
∗
= P
∗
(µ, P
1
, P
2
, . . . , P
q
, ε)
{µ}

{µ
n
: n ≥ m} ⊂ P
∗
⊂ O(µ, U
1

, U
2
, . . . , U
q
, ε).
P
∗
= P
∗
(µ, P
1
, P
2
, . . . , P
q
, ε) ∈ P
∗
.
P wcs
∗
X P
∗
wcs
∗
P
k
(X)
[1] V. V. Fedorchuk, Soviet
Math. Dokl. , 1986, pp. 1329 -1333.
[2] Ta Khac Cu, VNU,

Journal of science, , 2003, pp. 22 - 33.
[3] G. Gruenhage, E. Michael and Y. Tanaka,
Pacific J. Math., , 1984, pp. 303 - 332.
[4] Y. Tanaka, k , Topology Proc., , 1987, pp.
327 - 349.
[5] S. Liu and C. Liu, cs Topology Appl., ,
1996, pp. 51 - 60.
[6] Y.Ge, Publication de L’institute Mathema-
tique, Nouvelle serie, , 2003, pp. 121 - 128.
[7] Y. Ikeda and Y. Tanaka, k Topology Proc.,
, 1993, pp. 107 - 132.
P
k
(X)
X P
k
(X)