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

Probability measure functors preservingsome topological properties Ta Khac Cua, Luong Quoc Tuyenb Nguyen Thi Thu Hac, Le Thi Ngocc Abstract.. Fedorchuk introduced the concept of probabil

Probability measure functors preserving

some topological properties

Ta Khac Cu(a), Luong Quoc Tuyen(b) Nguyen Thi Thu Ha(c), Le Thi Ngoc(c)

Abstract In this paper, we introduce a topology inPk(X), and prove that ifX

have some networks, thenPk(X)so have.

1 Introduction and notions

In [1] V V Fedorchuk introduced the concept of probability measure functors with finite supports and proved that these functors preserve some topological invariants

In this paper, we study the action of probability measure functors on space with

probability measure functors

(1) suppµ = {x ∈ X : µ(x) > 0}is finite;

x∈suppµ

µ(x) = 1

µ =

q

X

i=1

miδxiwithq 6 k, whereδx(y)is Dirac function, that isδx(y) =

(

0 if y 6= x

1 if y = x,

and mi = µ(xi) > 0,

q

X

i=1

mi = 1 mi = µ(xi) is called the mass of µat xi In [1]

Letµ ∈ Pk(X),µ =Pq

i=1miδxi A neighborhood ofµis a set in the form

O(µ, U1, U2, , Uq, ε) = {µ ∈ Pk(X) : µ =

q+1

X

i=1

µi,suppµi ⊂ Ui, |k µi k −mi |< ε, i = 1, 2, , q + 1, mq+1 = 0},

1 NhËn bµi ngµy 08/5/2007 Söa ch÷a xong ngµy 20/10/2007.

where ε > 0, U1, U2, , Uq are disjoint neighborhoods of x1, x2, , xq, respectively,

Uq+1 = X\Sq

x∈suppµT Ui

µ(x)

k µi k6= 1

Pk(X); Ui ∈ B, i = 1, 2, , q; q 6 k, ε > 0}forms a base of a topology onPk(X) This

q

X

i=1

miδxi ∈ Pk(X), µn =

q

X

i=1

mniδxn

i, q 6 k The sequence

{µn} is called convergent toµo iff for ε > 0, there exists N > 0 such that for every

n ≥ N we havexn

i ∈ Ui, i = 1, , qand|mn

i − mo

i| < ε, i = 1, , q

k ∈ Nsuch thatxn∈ P forn ≥ k

(1) P is called a network forX, if wheneverx ∈ X, and any open neighborhoodU

(2) P is called acs-network forX, if whenever sequence{xn}converging tox ∈ X

{x}S{xn: n ≥ m} ⊂ P ⊂ U

(3) P is called acs∗-network forX, if whenever a sequence{xn} ⊂ X, converging

tox ∈ X, and neighborhoodU ofxthere exist a subsequence{xni : i ∈ N}of

{xn}andP ∈ P such that{x}S{xn i : i ∈ N} ⊂ P ⊂ U

(4) P is called a wcs∗-network for X, if whenever a sequence {xn} converging to

x ∈ X, and neighborhoodU ofxthere exist a subsequence {xni}of{xn}and

P ∈ P such that{xn i : i ∈ N} ⊂ P ⊂ U

x∈X

(a)Px is a network atx

(b) IfP1,P2 ∈ Px, then there existsP ∈ Pxsuch thatP ⊂ P1∩ P2

(1) Pis a weak base ofX, if forG ⊂ X,Gis open inXif and only if for everyx ∈ G

base atx

(2) P is ansn-network forX, if every element ofPxis a sequential neighborhood

2 The main results

q

X

i=1

miδxi ∈ Pk(X), q 6 k be a probability measure,P be

a network forX,Pi ∈ P, i = 1, 2, q, andε > 0 We define the setPµ∗ as follows

Pµ∗ = P∗(µ, P1, P2, , Pq, ε) =

=

(

µ =

q+1

X

i=1

µi :suppµi ⊂ Pi, |k µi k −mi |< ε, i = 1, 2, , q; Pq+1 = X \

q

[

i=1

Pi,

mq+1 = 0; Pi\Pj = φ with i 6= j, and k µi k= X

x i ∈suppµ T P i

µ(xi), k µq+1 k< ε

)

Denote

P∗

µ = {P∗(µ, P1, P2, , Pq, ε) : Pi ∈ P, i = 1, , q, ε > 0}, andP∗ = [

µ∈Pk(X)

P∗

µ

ThenP∗ is a cover ofPk(X)

Proof Letµbe inPk(X),µ =

q

X

i=1

miδxi ∈ Pk(X), q 6 k, andO(µ, U1, U2, , Uq, ε)

µ ∈ Pµ∗ ⊂ O(µ, U1, , Uq, ε)

Assume thatUi ∩ Uj = ∅ifi 6= j SinceP is a network, for everyi = 1, , qthere existsPi ∈ P such thatxi ∈ Pi ⊂ Ui,i = 1, , q Denote Pµ∗ = P∗(µ, P1, , Pq, ε) Then we haveµ ∈ Pµ∗ ⊂ O(µ, U1, , Uq, ε) Therefore,P∗ is the network forPk(X)

x∈X

conditions

(1) Px is a network at x;

(2) IfP1, P2 ∈ Px, there existsP ∈ Px such thatP ⊂ P1T P2

µ∈P k (X)

Pµ∗ so is

Proof (1) Similar to the proof of Theorem 2.2

(2) Suppose P1∗, P2∗ ∈ P∗

µ with P1∗ = P∗(µ, P11, , Pq1, ε1) where ε1 > 0, P2∗ =

P∗(µ, P2

1, , P2

q, ε2), where ε2 > 0, andµ =

q

X

i=1

miδxi ∈ Pk(X), q 6 k

i T P1

j = φwith i 6= j,P1

i ∈ Pxi, i = 1, 2, , q, and

Pi2T P2

j = φ with i 6= j, Pi2 ∈ Pxi, i = 1, 2, , q Without loss of generality we

i ∩ P2

i 6= ∅,i = 1, , q By assumption ofP we infer that there existsP3

i ⊂ P1

i T P2

i,i = 1, , q Chooseε < min{ε1, ε2}, and

P3∗ = P∗(µ, P13, , Pq3, ε) Then it follows thatP3∗ ⊂ P∗

1 T P∗

2

The Theorem 2.3 is proved

P∗ = P∗(µ, P1, , Pq, ε) ∈ Pµ∗ is a sequential neighborhood ofµ Letµ =

q

X

i=1

miδxi

wheremi = µ(xi) > 0,δxi is Dirac function,i = 1, , q, and{µn} be a convergent

q+1

X

i=1

µni Sinceµn −→ µ, we infer that

i ⊂ Pi, |k µn

i k −mi |< ε, i =

1, 2, , q; and k µn

q+1 k< εfor every n ≥ ni Put no = max{n1, n2, , nq} Then for every n ≥ no we have µn ∈ P∗(µ, P1, P2, , Pq, ε) Thus, P∗ is a sequential

Proof Let{µn}be a convergent sequence converging toµinPk(X),O(µ, U1, , Uq, ε)

i=1miδx i,µn =Pq+1

i=1µni with

q 6 k

µn∈ O(µ, U1, , Uq, ε), for every n ≥ m

Then for every n ≥ m, we have suppµni ⊂ Ui, |k µni k −mi |< ε, i = 1, 2, , q, and

k µn

q+1 k< ε Now, for everyi = 1, , qandn ≥ mwe choosexn

i ∈ suppµn

µn −→ µ, by definition of Fedorchuk topology it follows that {xn

xn

i −→ xi,i = 1, , q, it follows that for i = 1, , q and for neighborhoodUi ofxi

i }of{xn

i}such that{xnj

i }S{xi} ⊂ Pi ⊂ Ui Put P∗ = P∗(µ, P1, P2, , Pq, ε), and for every nj we consider µn j = Pq+1

i=1µnj

i

{µn j}[{µ} ⊂ P∗ = P∗(µ, P1, P2, , Pq, ε) ⊂ O(µ, U1, U2, , Uq, ε)

Proof Let{µn}be a convergent sequence converging toµinPk(X),O(µ, U1, , Uq, ε)

i=1miδx i,µn =Pq+1

i=1µni with

q 6 k

µn∈ O(µ, U1, , Uq, ε), for every n ≥ m

Then for every n ≥ m, we have suppµni ⊂ Ui, |k µni k −mi |< ε, i = 1, 2, , q, and

k µn

q+1 k< ε Now, for everyi = 1, , qandn ≥ mwe choosexn

i ∈ suppµn

sequence converging toxi for everyi = 1, , q

i −→ xi,i = 1, , q, it follows that for every

i = 1, , q and for neighborhoodUi of there existPi ∈ P and ami ∈ Nsuch that

{xi}[{xn

i : n ≥ mi} ⊂ Pi ⊂ Ui

Putm = max{m1, m2, , mq}, andP∗ = P∗(µ, P1, P2, , Pq, ε) This implies that

{µ}[{µn: n ≥ m} ⊂ P∗ ⊂ O(µ, U1, U2, , Uq, ε)

whereP∗ = P∗(µ, P1, P2, , Pq, ε) ∈ P∗

too

Proof Similar to the proof of Theorem 2.5

[1] V V Fedorchuk, Probability measure and absolute neighborhood retracts, Soviet Math Dokl 22, 1986, pp 1329 -1333

[2] Ta Khac Cu,Probability measure with finite supports on topological spaces, VNU, Journal of science, 29 (1), 2003, pp 22 - 33

[3] G Gruenhage, E Michael and Y Tanaka, Space determined by point-countable

[4] Y Tanaka, Point-countable cover and k-network, Topology Proc., 12, 1987, pp

327 - 349

[5] S Liu and C Liu, n spaces with point-countablecs-networks,Topology Appl.,74,

1996, pp 51 - 60

[6] Y.Ge, Characterization of metrizable spaces,Publication de L’institute Mathema-tique, Nouvelle serie, 74 (88), 2003, pp 121 - 128

[7] Y Ikeda and Y Tanaka,Spaces having star-countablek-networks,Topology Proc.,

18, 1993, pp 107 - 132

Tóm tắt

Hàm tử độ đo xác suất bảo toàn một số tính chất tôpô nào đó

(a) Department of Mathematics, Vinh University

(b) Department of Mathematics, Da Nang Pedagogical University

(c) Department of Post Graduate, Vinh University.