ZHIYUN YIN
Received 2 August 2005; Revised 29 June 2006; Accepted 9 July 2006
We give some characterizations of weak-open compact images of metric spaces.
Copyright © 2006 Hindawi Publishing Corporation. All rights reserved.
1. Introduction and definitions
To find internal characterizations of certain images of metric spaces is one of central problems in general topology. Arhangel’ski˘ı [1] showed that a space is an open compact image of a metric space if and only if it has a development consisting of point-finite open covers, and some characterizations for certain quotient compact images of metric spaces are obtained in [3,5,8]. Recently, Xia [12] introduced the concept of weak-open map- pings. By using it, certaing-first countable spaces are characterized as images of metric spaces under various weak-open mappings. Furthermore, Li and Lin in [4] proved that a space isg-metrizable if and only if it is a weak-openσ-image of a metric space.
The purpose of this paper is to give some characterizations of weak-open compact images of metric spaces, which showed that a space is a weak-open compact image of a metric space if and only if it has a weak development consisting of point-finitecs-covers.
In this paper, all spaces are Hausdorff, all mappings are continuous and surjective.N denotes the set of all natural numbers.τ(X) denotes the topology on a spaceX. For the usual product spacei∈NXi,πidenotes the projectioni∈NXiontoXi. For a sequence {xn}inX, denotexn = {xn:n∈N}.
Definition 1.1 [1]. Letᏼ=
{ᏼx:x∈X}be a collection of subsets of a spaceX.ᏼis called a weak base forXif
(1) for eachx∈X,ᏼxis a network ofxinX,
(2) ifU,V∈ᏼx, thenW⊂U∩Vfor someW∈ᏼx,
(3)G⊂X is open inX if and only if for eachx∈G, there existsP∈ᏼx such that P⊂G.
ᏼxis called a weak neighborhood base ofxinX, every element ofᏼxis called a weak neighborhood ofxinX.
Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences Volume 2006, Article ID 87840, Pages1–5
DOI 10.1155/IJMMS/2006/87840
Definition 1.2. Let f :X→Ybe a mapping.
(1) f is called a weak-open mapping [12], if there exists a weak baseᏮ= ∪{Ꮾy:y∈ Y}forY, and for each y∈Y, there existsxy∈ f−1(y) satisfying the following condition: for each open neighborhoodUofxy,By⊂ f(U) for someBy∈Ꮾy. (2) f is called a compact mapping, if f−1(y) is compact inXfor eachy∈Y. It is easy to check that a weak-open mapping is quotient.
Definition 1.3 [2]. LetXbe a space, andP⊂X. Then the following hold.
(1) A sequence{xn}inXis called eventually inP, if the{xn}converges tox, and there existsm∈Nsuch that{x} ∪ {xn:n≥m} ⊂P.
(2)Pis called a sequential neighborhood ofxinX, if whenever a sequence{xn}inX converges tox, then{xn}is eventually inP.
(3)Pis called sequential open inX, ifP is a sequential neighborhood at each of its points.
(4)Xis called a sequential space, if any sequential open subset ofXis open inX.
Definition 1.4 [7]. Letᏼbe a cover of a spaceX.
(1)ᏼis called acs-cover forX, if every convergent sequence inXis eventually in some element ofᏼ.
(2)ᏼis called ansn-cover forX, if every element ofᏼis a sequential neighborhood of some point inX, and for anyx∈X, there exists a sequential neighborhoodP ofxinXsuch thatP∈ᏼ.
Definition 1.5 [7]. Let{ᏼn}be a sequence of covers of a spaceX.
(1){ᏼn}is called a point-star network forX, if for eachx∈X,st(x,ᏼn)is a net- work ofxinX.
(2){ᏼn}is called a weak development forX, if for eachx∈X,st(x,ᏼn)is a weak neighborhood base forX.
2. Results
Theorem 2.1. The following are equivalent for a spaceX.
(1)Xis a weak-open compact image of a metric space.
(2)Xhas a weak development consisting of point-finitecs-covers.
(3)Xhas a weak development consisting of point-finitesn-covers.
Proof. (1)⇒(2). Suppose that f :M→X is a weak-open compact mapping with M a metric space. Let{ᐁn}be a sequence consisting of locally finite open covers ofM such thatᐁn+1is a refinement ofᐁnandst(K,ᐁn)forms a neighborhood base ofKinMfor each compact subsetKofM(see [7, Theorem 1.3.1]). For eachn∈N, putᏼn= f(ᐁn).
Since f is compact, then{ᏼn}is a point-finite cover sequence ofX.
Ifx∈V withV open inX, then f−1(x)⊂f−1(V). Since f−1(x) compact inM, then st(f−1(x),ᐁn)⊂ f−1(V) for somen∈N, and sost(x,ᏼn)⊂V. Hencest(x,ᏼn)forms a network ofxinX. Therefore,{ᏼn}is a point-star network forX.
We will prove that everyᏼk is acs-cover forX. Since f is weak-open, there exists a weak base Ꮾ= ∪{Ꮾx:x∈X} for X, and for eachx∈X, there exists mx∈ f−1(x)
satisfying the following condition: for each open neighborhoodUofmxinM,B⊂f(U) for someB∈Ꮾx.
For eachx∈Xandk∈N, let{xn}be a sequence converging to a pointx∈X. Take U∈ᐁk withmx∈U. ThenB⊂ f(U) for some B∈Ꮾx. Since Bis a weak neighbor- hood ofxinX, thenBis a sequential neighborhood ofxinXby [6, Corollary 1.6.18], so f(U)∈ᏼk is also. Thus{xn} is eventually in f(U). This implies that eachᏼk is a cs-cover forX. Since f(U) is a sequential neighborhood ofxinX, thenst(x,ᏼk) is also.
Obviously,Xis a sequential space. Sost(x,ᏼk)is a weak neighborhood base ofxinX.
In words,{ᏼn}is a weak development consisting of point-finitecs-covers forX.
(2)⇒(3). By Theorem A in [5],Xis a sequential space. It suffices to prove that ifᏼis a point-finitecs-cover forX, then some subset ofᏼis ansn-cover forX. For eachx∈X, denote (ᏼ)x= {Pi:i≤k}, where (ᏼ)x= {P∈ᏼ:x∈P}. If each element of (ᏼ)x is not a sequential neighborhood ofx inX, then for eachi≤k, there exists a sequence{xin} converging tox such that{xin}is not eventually inPi. For eachn∈N andi≤k, put yi+(n−1)k=xin, then{ym}converges toxand is not eventually in eachPi, a contradiction.
Thus there existsPx∈ᏼsuch thatPxis a sequential neighborhood ofxinX. PutᏲ= {Px:x∈X}, thenᏲis ansn-cover forX.
(3)⇒(1). Suppose{ᏼn}is a weak development consisting of point-finitesn-covers for X. For eachi∈N, letᏼi= {Pα:α∈Λi}, endowΛiwith the discrete topology, thenΛiis a metric space. Put
M=
α=(αi)∈
i∈N
Λi:Pαi
forms a network at some pointxαinX
, (2.1)
and endowMwith the subspace topology induced from the usual product topology of the collection{Λi:i∈N}of metric spaces, thenMis a metric space. SinceXis Hausdorff, xαis unique in X. For eachα∈M, we define f :M→X by f(α)=xα. For eachx∈X andi∈N, there existsαi∈Λisuch thatx∈Pαi. From{ᏼi}being a point-star network forX,{Pαi:i∈N}is a network ofxinX. Putα=(αi), thenα∈Mand f(α)=x. Thus f is surjective. Supposeα=(αi)∈Mand f(α)=x∈U∈τ(X), then there existsn∈N such thatPαn⊂U. Put
V= β∈M: thenth coordinate ofβisαn
. (2.2)
Thenα∈V∈τ(M), and f(V)⊂Pαn⊂U. Hence f is continuous.
For eachx∈Xandi∈N, put
Bi= αi∈Λi:x∈Pαi
, (2.3)
theni∈NBi is compact ini∈NΛi. Ifα=(αi)∈
i∈NBi, then Pαi is a network of x in X. Soα∈M and f(α)=x. Hence i∈NBi⊂ f−1(x); If α=(αi)∈ f−1(x), then x∈
i∈NPαi, soα∈
i∈NBi. Thus f−1(x)⊂
i∈NBi. Therefore, f−1(x)=
i∈NBi. This implies that f is a compact mapping.
We will prove that f is weak-open. For eachx∈X, since everyᏼiis ansn-cover forX, then there existsαi∈Λisuch thatPαiis a sequential neighborhood ofxinX. From{ᏼi} a point-star network forX,Pαiis a network ofxinX. Putβx=(αi)∈
i∈NΛi, then βx∈f−1(x).
Let{Umβx}be a decreasing neighborhood base ofβxinM, and put Ꮾx= fUmβx
:m∈N ,
Ꮾ= Ꮾx:x∈X, (2.4)
thenᏮsatisfies (1), (2) inDefinition 1.1. SupposeGis open inX. For eachx∈G, from βx∈f−1(x), f−1(G) is an open neighborhood ofβxinM. ThusUmβx⊂f−1(G) for some m∈N, so f(Umβx)⊂Gand f(Umβx)∈Ꮾx. On the other hand, supposeG⊂X and for x∈G, there existsB∈Ꮾx such that B⊂G. LetB= f(Umβx) for somem∈N, and let {xn}be a sequence converging toxinX. SincePαi is a sequential neighborhood ofxin Xfor eachi∈N, then{xn}is eventually inPαi. For eachn∈N, ifxn∈Pαi, letαin=αi; if xn∈Pαi, pickαin∈Λisuch thatxn∈Pαin. Thus there existsni∈Nsuch thatαin=αifor alln > ni. So{αin}converges toαi. For eachn∈N, put
βn=(αin)∈
i∈N
Λi, (2.5)
then f(βn)=xnand{βn}converges toβx. Since Umβx is an open neighborhoodβx in M, then{βn}is eventually inUmβx, so{xn}is eventually inG. HenceGis a sequential neighborhood ofx. SoGis sequential open inX. ByXbeing a sequential space,Gis open inX. This impliesᏮis a weak base forX.
By the idea ofᏮ, f is weak-open.
We give examples illustratingTheorem 2.1of this note.
Example 2.2. LetX be the Arens spaceS2 (see [6, Example 1.8.6]). It is not difficult to see that the space is a weak-open compact image of a metric space. ButXis not an open compact image of a metric space, becauseXis not developable. Thus the following holds.
A weak-open compact image of a metric space is not always an open compact image of a metric space.
Example 2.3. LetYbe the weak Cauchy space in [10, Example 2.14(3)]. By the construc- tion,Y is a quotient compact image of a metric space. ButY is not Cauchy,Y is not a weak-open compact image of a metric space byTheorem 2.1. Thus the following holds:
A quotient compact image of a metric space is not always a weak-open compact image of a metric space.
Acknowledgments
This work is supported by the NSF of Hunan Province in China (no. 05JJ40103) and the NSF of Education Department of Hunan Province in China (no. 03C204).
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Zhiyun Yin: Department of Information, Hunan Business College, Changsha, Hunan 410205, China
E-mail address:[email protected]
