# Onπ-Images of Locally Separable Metric Spaces

Authors: Tran Van An; Nguyen Vangiven-names Dung
Journal: International Journal of Mathematics and Mathematical Sciences
 (2007-11-26)

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## Abstract

We characterizeπ-images of locally separable metric spaces
by means of covers having π-property. As its application, we obtain characterizations
of compact-covering (sequence-covering, pseudo-sequence-covering, and sequentially quotient)
π-images of locally sparable metric spaces.


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## Body

## 1. Introduction

To determine what spaces are the images of “nice” spaces
under “nice” mappings is one of the central questions of general topology
in [1]. In the past, many noteworthy results on
images of metric spaces have been obtained. For a survey in this field,
see [2], for example. A characterization for
a quotient compact image of a locally separable metric space is obtained in
[3]. Also, such a quotient image is precisely
a pseudo-sequence-covering quotient compact image of
a locally separable metric space [4].
Recently, π-images
of metric spaces cause attention once again in [5, 6]. It is known that a space is
a compact-covering (resp., sequence-covering, pseudo-sequence-covering, sequentially-quotient) π-image of a metric space if and only if it has a point-star
network consisting of cfp-covers (resp.,
cs-covers, wcs-covers, cs*-covers) [5–7]. In a personal communication,
the first author of [6] informs that it seems to
be difficult to obtain “nice” characterizations of “nice” images of
locally separable metric spaces (instead of metric or locally compact
metric domains). Thus, we are interested in the following question.Question 1.1.
How are compact-covering (resp., sequence-covering,
pseudo-sequence-covering, sequentially-quotient)π-images of locally sparable metric spaces characterized?In this paper, we characterize compact-covering
(resp., sequence-covering, pseudo-sequence-covering, sequentially-quotient)π-images
of locally separable metric spaces by means of cfp-covers (resp., cs-covers, wcs-covers, cs*-covers) for compact subsets (resp., convergent sequences) in
a space and covers having π-property to answer
Question 1.1 completely. As applications
of these results, we get characterizations on quotient π-images of locally separable metric
spaces.Throughout this paper, all spaces are assumed to be
regular andT1, all mappings are assumed continuous and onto,
a convergent sequence includes its limit point, ℕ denotes the set
of all natural numbers. Let f:X→Y be a mapping, x∈X, and 𝒫 be a collection
of subsets of X, we denote st(x,𝒫)=⋃{P∈𝒫:x∈P}, 𝒫x={P∈𝒫:x∈P}, ⋃𝒫=⋃{P:P∈𝒫}, and f(𝒫)={f(P):P∈𝒫}. We say that a convergent sequence {xn:n∈ℕ}∪{x} converging to x is eventually (resp., frequently) in A if {xn:n≥n0}∪{x}⊂A for some n0∈ℕ (resp., {xnk:k∈ℕ}∪{x}⊂A for some
subsequence {xnk:k∈ℕ} of {xn:n∈ℕ}). For terms
which are not defined here, please refer to [2, 8].Let𝒫 be a collection
of subsets of a space X, and K be a subset of X.𝒫 is a cover forKinX, if K⊂⋃𝒫. When K=X, a cover for K in X is a cover
ofX [8]. A cover 𝒫 for X is a compact cover if all members of 𝒫 are compact.For eachx∈X, 𝒫 is a network atx if x∈P for every P∈𝒫, and if x∈U with U open in X, there exists P∈𝒫 such that x∈P⊂U.𝒫 is a k-coverforKinX, if for each compact subset H of K there exists a
finite subfamily ℱ of 𝒫 such that H⊂⋃ℱ. When K=X, a k-cover for K in X is a k-coverforX.𝒫 is a cfp-coverforKinX, if for each compact subset H of K there exists a
finite subfamily ℱ of 𝒫 such that H⊂⋃{CF:F∈ℱ}, where CF is closed and CF⊂F for every F∈ℱ. Note that such an ℱ is a full
cover in the sense of [9]. When K=X, a cfp-cover for K in X is a cfp-coverforX [10].𝒫 is a cs-coverforKinX (resp., cs*-coverforKinX), if for each convergent sequence S in K, S is eventually
(resp., frequently) in some P∈𝒫. When K=X, a cs-cover for K in X (resp., cs*-cover for K in X) is a cs-coverforX [11] (resp., cs*-coverforX [12]), or a condition(c3) (resp., (c2)) [5].𝒫 is a wcs-coverforKinX if for each
convergent sequence S converging to x in K there exists a
finite subfamily ℱ of 𝒫x such that S is eventually
in ⋃ℱ. When K=X, a wcs-cover for K in X is a wcs-cover [7].It is clear that if𝒫 is a cover
(resp., k-cover, cfp-cover, cs-cover, wcs-cover, cs*-cover), then 𝒫 is a cover
(resp., k-cover, cfp-cover, cs-cover, wcs-cover, cs*-cover) for K in X.
Remark 1.2.
(1) Closedk-cover for K in X⇒cfp-cover for K in X⇒k-cover for K in X;
(2)cfp-cover for K in X, or cs-cover for K in X⇒wcs-cover for K in X⇒cs*-cover for K in X.For eachn∈ℕ, let 𝒫n be a cover for X. {𝒫n:n∈ℕ} is a refinement
sequence for X if 𝒫n+1 is a refinement
of 𝒫n for each n∈ℕ. A refinement sequence for X is a refinement of X in the sense of
[4].Let{𝒫n:n∈ℕ} be a refinement
sequence for X. {𝒫n:n∈ℕ} is a point-star
network for X, if {st(x,𝒫n):n∈ℕ} is a network at x for each x∈X. Note that this notion is used without the assumption
of a refinement sequence in [13], and in [5] ⋃{𝒫n:n∈ℕ} is a σ-strongnetwork for X.A cover{Xλ:λ∈Λ} for X is called to
have π-property if each Xλ has a
refinement sequence {𝒫λ,n:n∈ℕ} of countable
covers for Xλ, and for each x∈U with U open in X, there is n∈ℕ such that(1.1)⋃{st(x,𝒫λ,n):λ∈Λwithx∈Xλ}⊂U.Let{𝒫n:n∈ℕ} be a point-star
network for a space X. For every n∈ℕ, put 𝒫n={Pα:α∈An}, and An is endowed with
discrete topology. Put(1.2)M={a=(αn)∈∏n∈ℕAn:{Pαn:n∈ℕ}formsanetworkatsomepointxainX}.Then M, which is a subspace of the product space ∏n∈ℕAn, is a metric space with metric d described as
follows.Leta=(αn),b=(βn)∈M. If a=b, then d(a,b)=0. If a≠b, then d(a,b)=1/(min{n∈ℕ:αn≠βn}).Definef:M→X by choosing f(a)=xa, then f is a mapping,
and (f,M,X,{𝒫n}) is a Ponomarev's
system [14], and
if without the assumption of a refinement sequence in the notion of point-star
networks, then (f,M,X,{𝒫n}) is a Ponomarev's
system in the sense of [13].Letf:X→Y be a mapping.f is a compact-covering
mapping [15], if
every compact subset of Y is the image of
some compact subset of X.f is a sequence-covering
mapping [16], if
every convergent sequence of Y is the image of
some convergent sequence of X.f is a pseudo-sequence-covering
mapping [5], if
every convergent sequence of Y is the image of
some compact subset of X.f is a subsequence-covering
mapping [3], if
for every convergent sequence S of Y, there is a compact subset K of X such that f(K) is a
subsequence of S.f is a sequentially-quotient
mapping [17], if
for every convergent sequence S of Y, there is a convergent sequence L of X such that f(L) is a subsequence
of S.f is a pseudo-open
mapping [1], if y∈intf(U) whenever f−1(y)⊂U with U open in X.f is a π-mapping [1], if for every y∈Y and for every
neighborhood U of y in Y, d(f−1(y),X−f−1(U))>0, where X is a metric
space with a metric d.LetX be a space. We
recall that X is sequential [18], if a subset A of X is closed if
and only if any convergent sequence in A has a limit
point in A. Also, X is Fréchet (or Fréchet Urysohn) if for each x∈A¯, there exists a sequence in A converging to x.Note that, for a mappingf:X→Y, f is
compact-covering or sequence-covering ⇒f is
pseudo-sequence-covering ⇒f is
subsequence-covering [4]. Also, f is quotient if
and only if f is
subsequence-covering for Y being
sequential [3].


## 2. Main Results

Lemma 2.1.
Let𝒫 be a countable
cover for a convergent sequence S in a space X. Then the following are equivalent:
(1)
𝒫 is a cfp-cover for S in X;(2)
𝒫 is a wcs-cover for S in X;(3)
𝒫 is a cs*-cover for S in X.Proof.
(1)⇒ (2) ⇒ (3). By Remark
1.2.
(3)⇒ (1). Let H be a compact
subset of S. We can assume that H is a
subsequence of S. Since 𝒫 is countable,
put 𝒫x={Pn:n∈ℕ}, where x is the limit
point of S. Then H is eventually
in ⋃n≤kPn for some k∈ℕ. If not, then for any k∈ℕ, H is not
eventually in ⋃n≤kPn. So, for every k∈ℕ, there exists xnk∈S−⋃n≤kPn. We may assume n1<n2<⋯<nk−1<nk<nk+1<⋯. Put H′={xnk:k∈ℕ}∪{x}, then H′ is a
subsequence of S. Since 𝒫 is a cs*-cover for S in X, there exists m∈ℕ such that H′ is frequently
in Pm. This contradicts to the construction of H′. So H is eventually
in ⋃n≤kPn for some k∈ℕ. It implies that 𝒫 is a cfp-cover for S in X.The following lemma is routinely shown, so we omit the proof.Lemma 2.2.
Letf:X→Y be a mapping.
(1)
If𝒫 is a k-cover for a
compact set K in X, then f(𝒫) is a k-cover for f(K) in Y.(2)
If𝒫 is a cs-cover for a
convergent sequence S in X, then f(𝒫) is a cs-cover for f(S) in Y.Same as [6, Lemma 2.2(2)(ii)],
we get the following.Lemma 2.3.
Let(f,M,X,{𝒫n}) be a
Ponomarev's system. For a convergent sequence S of X, if 𝒫n is a cs-cover for S in X for each n∈ℕ, then there exists a convergent sequence L of M such that f(L)=S.Proof.
PutS={xi:i∈ℕ}∪{x}, where x is the limit
point. For each n∈ℕ, since 𝒫n is a cs-cover for S, S is eventually
in some Pαn∈𝒫n. For each i∈ℕ, if xi∈Pαn, let αn,i=αn; if xi∉Pαn, pick αn,i∈An such that xi∈Pαn,i. Thus there exists in∈ℕ such that αn,i=αn for all i>in. So {αn,i:i∈ℕ} converges to αn. For each i∈ℕ, put ai=(αn,i)∈∏n∈ℕAn, and put a=(αn)∈∏n∈ℕAn. Then ai∈f−1(xi), a∈f−1(x), and L={ai:i∈ℕ}∪{a} converges to a. It implies that L is a convergent
sequence of M and f(L)=S.Proposition 2.4.
The following are equivalent for a spaceX.
(1)
X is a π-image of a
locally separable metric space,(2)
X has a cover {Xλ:λ∈Λ} having π-property.Proof.
(1)⇒ (2). Let f:M→X be a π-mapping from a
locally separable metric space M with metric d onto X. Since M is a locally
separable metric space, M=⊕λ∈ΛMλ, where each Mλ is a separable
metric space by [8,
4.4.F]. For each λ∈Λ, let
Dλ be a countable dense subset of Mλ,
and put fλ=f|Mλ and Xλ=fλ(Mλ). For each a∈Mλ and n∈ℕ, put B(a,1/n)={b∈Mλ:d(a,b)<1/n}, ℬλ,n={B(a,1/n):a∈Dλ}, and 𝒫λ,n=fλ(ℬλ,n). It is clear that {𝒫λ,n:n∈ℕ} is a sequence
of countable covers for Xλ, and 𝒫λ,n+1 is a refinement
of 𝒫λ,n for every n∈ℕ. We will prove that π-property is
satisfied.
For eachx∈U with U open in X. Since f is a π-mapping, d(f−1(x),M−f−1(U))>2/n for some n∈ℕ. Then, for each λ∈Λ with x∈Xλ, we get d(fλ−1(x),Mλ−fλ−1(Uλ))>2/n, where Uλ=U∩Xλ. Let a∈Dλ and x∈fλ(B(a,1/n))∈𝒫λ,n. We will prove that B(a,1/n)⊂fλ−1(Uλ). In fact, if B(a,1/n)⊄fλ−1(Uλ), then pick b∈B(a,1/n)−fλ−1(Uλ). Note that fλ−1(x)∩B(a,1/n)≠∅, pick c∈fλ−1(x)∩B(a,1/n), then d(fλ−1(x),Mλ−fλ−1(Uλ))≤d(c,b)≤d(c,a)+d(a,b)<2/n. It is a contradiction. So B(a,1/n)⊂fλ−1(Uλ), thus fλ(B(a,1/n))⊂Uλ. Then st(x,𝒫λ,n)⊂Uλ. It implies that ⋃{st(x,𝒫λ,n):λ∈Λwithx∈Xλ}⊂U.
(2)⇒ (1). For each λ∈Λ, let x∈Uλ with Uλ open in Xλ. We get that Uλ=U∩Xλ with some U open in X. Since ⋃{st(x,𝒫λ,n):λ∈Λwithx∈Xλ}⊂U for some n∈ℕ, st(x,𝒫λ,n)⊂Uλ. It implies that {𝒫λ,n:n∈ℕ} is a point-star
network for Xλ. Then the Ponomarev's system (fλ,Mλ,Xλ,{𝒫λ,n}) exists. Since
each 𝒫λ,n is countable, Mλ is a separable
metric space with metric dλ described as
follows. For a=(αn),b=(βn)∈Mλ, if a=b, then dλ(a,b)=0, and if a≠b, then dλ(a,b)=1/(min{n∈ℕ:αn≠βn}).
PutM=⊕λ∈ΛMλ and define f:M→X by choosing f(a)=fλ(a) for every a∈Mλ with some λ∈Λ. Then f is a mapping
and M is a locally
separable metric space with metric d defined as
follows. For a,b∈M, if a,b∈Mλ for some λ∈Λ, then d(a,b)=dλ(a,b), and otherwise, d(a,b)=1.
We will prove thatf is a π-mapping. Let x∈U with U open in X, then ⋃{st(x,𝒫λ,n):λ∈Λwithx∈Xλ}⊂U for some n∈ℕ. So, for each λ∈Λ with x∈Xλ, we get st(x,𝒫λ,n)⊂Uλ, where Uλ=U∩Xλ. It is implies that dλ(fλ−1(x),Mλ−fλ−1(Uλ))≥1/n. In fact, if a=(αk)∈Mλ such that dλ(fλ−1(x),a)<1/n, then there is b=(βk)∈fλ−1(x) such that dλ(a,b)<1/n. So αk=βk if k≤n. Note that x∈Pβn⊂st(x,𝒫λ,n)⊂Uλ. Then fλ(a)∈Pαn=Pβn⊂st(x,𝒫λ,n)⊂Uλ. Hence a∈fλ−1(Uλ). It implies that dλ(fλ−1(x),a)≥1/n if a∈Mλ−fλ−1(Uλ). So dλ(fλ−1(x),Mλ−fλ−1(Uλ))≥1/n. Therefore(2.1)d(f−1(x),M−f−1(U))=inf{d(a,b):a∈f−1(x),b∈M−f−1(U)}=min{1,inf{dλ(a,b):a∈fλ−1(x),b∈Mλ−fλ−1(Uλ),λ∈Λ}}≥1/n>0.It implies that f is a π-mapping.Theorem 2.5.
The following are equivalent for a spaceX.
(1)
X is a
compact-covering π-image of a
locally separable metric space;(2)
X has a cover {Xλ:λ∈Λ} having π-property
satisfying that for each compact subset K of X, there is a finite subset ΛK of Λ such that K has a finite
compact cover {Kλ:λ∈ΛK}, and for each λ∈ΛK and n∈ℕ, 𝒫λ,n is a k-cover for Kλ in Xλ;(3)
same as (2), but replace the prefix “k-” by “cfp-”.Proof.
(1)⇒ (2). By using
notations and arguments in the proof (1) ⇒ (2) of
Proposition 2.4 again, X has a cover {Xλ:λ∈Λ} having π-property. For
each compact subset K of X, since f is
compact-covering, K=f(L) for some
compact subset L of M. By compactness of L, Lλ=L∩Mλ is compact and ΛK={λ∈Λ:Lλ≠∅} is finite. For
each λ∈ΛK, put Kλ=f(Lλ), then {Kλ:λ∈ΛK} is a finite
compact cover for K. For each n∈ℕ, since ℬλ,n is a k-cover for Lλ in Mλ, 𝒫λ,n is a k-cover for Kλ in Xλ by Lemma 2.2.
(2)⇒ (3). For each λ∈ΛK and n∈ℕ, since 𝒫λ,n is countable,
every member of 𝒫λ,n can be chosen
closed in Xλ. Thus, 𝒫λ,n is a cfp-cover for Kλ in Xλ by Remark 1.2.
(3)⇒ (1). By using
notations and arguments in the proof (2) ⇒ (1) of
Proposition 2.4 again, X is a π-image of a
locally separable metric space. It suffices to prove that the mapping f is
compact-covering.
For each compact subsetK of X, there is a finite subset ΛK of Λ such that K has a finite compact
cover {Kλ:λ∈ΛK}, and for each λ∈ΛK and n∈ℕ, 𝒫λ,n is a cfp-cover for Kλ in Xλ. It follows from
[13, Lemma 13] that Kλ=fλ(Lλ) with some
compact subset Lλ of Mλ. Put L=⋃{Lλ:λ∈ΛK}, then L is a compact
subset of M and f(L)=K. It implies that f is
compact-covering.Theorem 2.6.
The following are equivalent for a spaceX.
(1)
X is a
pseudo-sequence-covering π-image of a
locally separable metric space;(2)
X has a cover {Xλ:λ∈Λ} having π-property
satisfying that for each convergent sequence S of X, there is a finite subset ΛS of Λ such that S has a finite
compact cover {Sλ:λ∈ΛS}, and for each λ∈ΛS and n∈ℕ, 𝒫λ,n is a wcs-cover for Sλ in Xλ;(3)
same as (2),
but replace the prefix “wcs-” by “cs*-”.Proof.
Using Lemma2.1, notations
and arguments in the proof of Theorem 2.5, here “
pseudo-sequence-covering”
and “convergent sequence” play the roles of “
compact-covering” and “compact subset”, respectively.Theorem 2.7.
The following are equivalent for a spaceX.
(1)
X is a
sequence-covering π-image of a
locally separable metric space;(2)
X has a cover {Xλ:λ∈Λ} having π-property
satisfying that for each convergent sequence S of X, there is λ∈Λ such that S is eventually
in Xλ, and for each n∈ℕ, 𝒫λ,n is a cs-cover for S∩Xλ in Xλ.Proof.
(1)⇒ (2). By using
notations and arguments in the proof (1) ⇒ (2) of
Proposition 2.4 again, X has a cover {Xλ:λ∈Λ} having π-property. For
each convergent sequence S of X, since f is
sequence-covering, S=f(L) for some
convergent sequence L of M. Note that L is eventually
in some Mλ. Thus, S is eventually
in Xλ with some λ∈Λ. On the other hand, for each n∈ℕ, ℬλ,n is a cs-cover for L∩Mλ in Mλ. It follows from Lemma 2.2 that 𝒫λ,n is a cs-cover for f(L∩Mλ) in Xλ. Then 𝒫λ,n is a cs-cover for S∩Xλ in Xλ.
(2)⇒ (1). By using
notations and arguments in the proof (2) ⇒ (1) of
Proposition 2.4 again, X is a π-image of a
locally separable metric space. It suffices to prove that the mapping f is
sequence-covering.
For each convergent sequenceS of X, there is λ∈Λ such that S is eventually
in Xλ, and for each n∈ℕ, 𝒫λ,n is a cs-cover for S∩Xλ in Xλ. Since S∩Xλ is a convergent
sequence in Xλ, there is a convergent sequence Lλ in Mλ such that fλ(Lλ)=S∩Xλ by Lemma 2.3.
Note that S−Xλ=f(F) with some
finite subset F of M. Put L=F∪Lλ, then L is a convergent
sequence in M and f(L)=S. It implies that f is
sequence-covering.Theorem 2.8.
The following are equivalent for a
spaceX.
(1)
X is a
subsequence-covering π-image of a
locally separable metric space;(2)
X is a
sequentially-quotient π-image of a
locally separable metric space;(3)
X has a cover {Xλ:λ∈Λ} having π-property
satisfying that for each convergent sequence S of X, there is λ∈Λ such that 𝒫λ,n is a cs-cover for some
subsequence Sλ of S in Xλ for each n∈ℕ;(4)
same as (3), but replace the prefix “cs-” by “wcs-”, or “cs*-”.Proof.
(1)⇒ (2). By
[4, Proposition 2.1].
(2)⇒ (3). For each
convergent sequence S of X, since f is
sequentially-quotient, there is a convergent sequence L of M such that f(L) is a
subsequence of S. By the proof of (1) ⇒ (2) in Theorem
2.7, where f(L) plays the role
of S in the
argument, we get (3).
(3)⇒ (4). By Lemma
2.1, “wcs-” and “cs*-” are
equivalent. It is routine when “cs-” is replaced
by “cs*-”.
(4)⇒ (1). It
suffices to assume that 𝒫λ,n is a cs*-cover for a
subsequence T of S in Xλ. By the proof of (3) ⇒ (1) in Theorem
2.5, where T plays the role
of compact subset K in the
argument, we get (1).Based on above results, it is easy to get
characterizations for quotientπ-images of
locally separable metric spaces as follows.Corollary 2.9.
The following are equivalent for a spaceX.
(1)
X is a
compact-covering quotient π-image of a
locally separable metric space;(2)
X is a sequential
space satisfying (2), or (3) in Theorem 2.5.Corollary 2.10.
The following are equivalent for a spaceX.
(1)
X is a
pseudo-sequence-covering quotient π-image of a
locally separable metric space;(2)
X is a sequential
space satisfying (2), or (3) in Theorem 2.6.Corollary 2.11.
The following are equivalent for a spaceX.
(1)
X is a
sequence-covering quotient π-image of a
locally separable metric space;(2)
X is a sequential
space satisfying (2) in Theorem 2.7.Corollary 2.12.
The following are equivalent for a spaceX.
(1)
X is a quotient π-image of a
locally separable metric space;(2)
X is a sequential
space satisfying (3), or (4) in Theorem 2.8.Remark 2.13.
“Quotient” and “sequential” in
above corollaries can be replaced by “pseudo-open”
and “Fréchet”, respectively.In [6], the authors raised a question whether
pseudo-sequence-covering quotient π-images of
metric spaces and quotient π-images of
metric spaces are equivalent. Similar to this question we have the
following.Question 2.14.
Are pseudo-sequence-covering
quotientπ-images of
locally separable metric spaces and quotient π-images of
locally separable metric spaces equivalent?



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