ARCHIVUM MATHEMATICUM (BRNO) Tomus 42 (2006), 135 – 140
SPACES WITH σ-LOCALLY COUNTABLE WEAK-BASES
ZHAOWEN LI
Abstract. In this paper, spaces withσ-locally countable weak-bases are characterized as the weakly open msss-images of metric spaces (or g-first countable spaces withσ-locally countablecs-networks).
To find the internal characterizations of certain images of metric spaces is an interesting research topic on general topology. Recently, S. Xia[12]introduced the concept of weakly open mappings, by using it, certain g-first countable spaces are characterized as images of metric spaces under various weakly open mappings.
The present paper establish the relationships spaces withσ-locally countable weak- bases and metric spaces by means of weakly pen mappings and msss-mappings, and give a characterization of spaces withσ-locally countable weak-bases.
In this paper, all spaces are regular andT1, all mappings are continuous and surjective. N denotes the set of all natural numbers. ω denotes N ∪ {0}. For a family P of subsets of a space X and a mapping f : X →Y, denote f(P) = {f(P) : P ∈ P}. For the usual product space Q
i∈N
Xi, pi denotes the projection from Q
i∈N
Xi ontoXi.
Definition 1. Let P = ∪{Px : x ∈ X} be a family of subsets of a space X satisfying that for eachx∈X,
(1) Px is a network ofxinX,
(2) IfU, V ∈ Px, thenW ⊂U∩V for someW ∈ Px.
P is called a weak-base for X[1] ifG⊂X is open inX if and only if for each x∈G, there existsP ∈ Px such thatP ⊂G.
A spaceX is calledg-first countable[1]ifX has a weak-baseP such that each Pxis countable.
A spaceX is called ag-metrizable space[4]ifX has aσ-locally finite weak-base.
Definition 2. LetP be a cover of a spaceX.
2000Mathematics Subject Classification: 54E99, 54C10.
Key words and phrases: weak-bases,cs-networks,k-networks,g-first countable spaces, weakly open mappings, msss-mappings.
This work is supported by the NNSF of China (No.10471020, 10471035) and the NSF of of Hunan Province in China (No. 04JJ6028).
Received September 21, 2004.
(1) Pis called ak-network forXif for each compact subsetKofXand its open neighbourhoodV, there exists a finite subfamilyP′ofP such thatK⊂ ∪P′ ⊂V. (2) P is called acs-network forX if for eachx∈X, its open neighbourhood V and a sequence{xn} converging tox, there existsP ∈ P such that{xn :n≥ m} ∪ {x} ⊂P⊂V for some m∈N.
A spaceX is called anℵ-space ifX has aσ-locally finite k-network.
Definition 3. Letf :X →Y be a mapping.
(1) f is called a weakly open mapping[12] if there exists a weak-base B =
∪{By :y∈Y}forY and fory ∈Y, there existsx(y)∈f−1(y) satisfying condition (∗): for each open neighbourhoodU of x(y),By⊂f(U) for someBy∈ By.
(2) f is called a msss-mapping[7](i.e., metrizably stratified strongs-mapping) if there exists a subspaceX of the usual product space Q
i∈N
Xiof the family{Xi: i∈N} of metric spaces satisfying the following condition: for eachy ∈Y, there exists an open neighbourhood sequence{Vi}ofyinY such that eachpif−1(Vi) is separable inXi.
Theorem 4. A space Y has a σ-locally countable weak-base if and only if Y is the weakly open msss-image of a metric space.
Proof. Sufficiency.SupposeY is the image of a metric spaceX under a weakly open msss-mappingf. Sincef is a msss-mapping, then exists a family{Xi:i∈N} of metric spaces satisfying the condition of Definition 3 (2).
For eachi∈N, letPi be aσ-locally finite base forXi, put Bi=n
X∩ \
j≤i
p−1j (Pj)
:Pj∈ Pj andj≤io ,
B=∪{Bi:i∈N}.
ThenBis a base forX. For eachn∈N, put V =\
j≤i
Vi,
then{Q∈f(Bi) :V ∩Q6= Φ}is countable. Thusf(Bi) is locally countable inY. Hencef(B) isσ-locally countable inY.
Sincef is a weakly open mapping, then exists a weak-baseP =∪{Py :y∈Y} forY such that for eachy∈Y, there existsx(y)∈f−1(y) satisfying the condition (∗) of Definition 3 (1). For eachy∈Y, put
Fi,y={f(B) :x(y)∈B ∈ Bi}, Fy =∪{Fi,y:i∈N},
Fi=∪{Fi,y:y∈Y}, F =∪{Fy:y∈Y}.
Obviously, Fi ∈ f(Bi) for eachi ∈ N, then Fi is locally countable in Y. Thus F=∪{Fi:i∈N}isσ-locally countable inY. We will prove thatFis a weak-base forY.
It is obvious thatF satisfies the condition (1) of Definition 1. For eachy∈Y, supposeU, V ∈ Fy, thenU ∈ Fm,y, V ∈ Fn,yfor somem, n∈N. Thus there exist B1∈ BmandB2∈ Bnsuch thatx(y)∈B1∩B2,f(B1) =U andf(B2) =V. Since B1, B2 ∈ B and Bis a base forX, then there exist l∈ N andB ∈ Bl such that x(y)∈B ⊂B1∩B2. Thusf(B)∈ Fl,y ⊂ Fy andf(B)⊂f(B1∩B2)⊂U ∩V. HenceF satisfies the condition (2) of Definition 1.
SupposeG⊂Y and fory∈G, there existsF ∈ Fysuch thatF ⊂G, then there existsB∈ Bsuch thatx(y)∈BandF =f(B). SinceBis an open neighbourhood ofx(y) andf is a weakly open mapping, then existsPy∈ Pysuch thatPy⊂f(B).
Thus for each y ∈ G, there exists Py ∈ Py such thatPy ⊂G. HenceGis open in Y because P is a weak-base for Y. On the other hard. Suppose G ⊂ Y is open in Y, then for eachy ∈ G, x(y) ∈ f−1(G). Since B is a base for X, then x(y)∈B ⊂f−1(G) for someB∈ B. Thusf(B)∈ Fy andf(B)⊂G.
ThereforeF is a weak-base forY.
Necessity. SupposeY has aσ-locally countable weak-base. LetP =∪{Pi:i ∈ N} be a σ-locally countable weak-base for Y, where each Pi ={Pα: α∈Ai} is a locally countable of subsets ofY which is closed under finite intersections and Y ∈ Pi ⊂ Pi+1. For each i∈ N, endow Ai with discrete topology, then Ai is a metric space. Put
X =n
α= (αi)∈ Y
i∈N
Ai :{Pαi:i∈N} ⊂P
forms a network at some pointx(α)∈Xo ,
and endowXwith the subspace topology induced from the usual product topology of the family {Ai :i ∈ N} of metric spaces, then X is a metric space. Since Y is Hausdroff, x(α) is unique in Y for each α ∈ X. We define f : X → Y by f(α) =x(α) for eachα∈X. BecauseP is aσ-locally countable weak-base forY, thenf is surjective. For eachα= (αi)∈M ,f(α) =x(α). SupposeV is an open neighbourhood of x(α) in Y, there exists n∈N such thatx(α)∈Pαn ⊂V, set W ={c∈X : the n-the coordinate ofcisαn}, thenW is an open neighbourhood ofαin X, and f(W)⊂Pαn ⊂V. Hencef is continuous. We will show thatf is a weakly open msss-mapping.
(i) f is a msss-mapping. For eachx∈X and eachi∈N, there exists an open neighbourhoodVi ofxin X such that {α∈Ai:Pα∩Vi6= Φ}is countable. Put
Bi={α∈Ai:Pα∩Vi6= Φ},
then pif−1(Vi) ⊂ Bi. Thus pif−1(Vi) is separable in Ai, Hence f is a msss- mapping.
(ii) f is a weakly open mapping For eachn∈N andαn∈An, put
V(α1,· · ·, αn) ={β∈X : for eachi≤n, the i-th coordinate ofβ isαi}. It is easy to check that {V(α1,· · ·, αn) :n∈N}is a locally neighbourhood base ofαinX.
Claim. f V(α1,· · ·, αn)
= T
i≤n
Pαi for eachn∈N.
For each i ≤ n, f V(α1,· · · , αn)
⊂ Pαi, then f V(α1,· · ·, αn)
⊂ T
i≤n
Pαi. On the other hand. For each x ∈ T
i≤n
Pαi, there is β = (βj) ∈ X such that f(β) = x. For each j ∈N, Pβj ∈ Pj ⊂ Pj+n, then there is αj+n ∈ Aj+n such that Pαj+n =Pβj. Set α = (αj), then α∈ V(α1,· · ·, αn) andf(α) =x. Thus
T
i≤n
Pαi⊂f V(α1,· · ·, αn)
. Hencef V(α1,· · ·, αn)
= T
i≤n
Pαi. DenotePy={P ∈ P :y∈P}, thenP=∪{Py :y∈Y}.
For eachy ∈Y, by the idea P, there exists (αi)∈ Q
i∈N
Ai such that {Pαi :i∈ N} ⊂ P is a network ofy inY, then α= (αi)∈f−1(y).
Suppose G is an open neighbourhood of α in X, then there exists j ∈ N such that V(α1,· · ·, αj) ⊂ G. Thus f V(α1,· · · , αj)
⊂ f(G). By the Claim, f V(α1,· · ·, αj)
= T
i≤j
Pαi. Since Py ⊂ T
i≤j
Pαi for some Py ∈ Py. Hence Py ⊂ f(G).
Therefore there exists a weak-base P for Y and α ∈ f−1(y) satisfying the condition (∗) of Definition 3 (1), and sof is a weakly open mapping.
Theorem 5. For a space X,(1) ⇐⇒ (2)⇒(3)below hold.
(1)X has a σ-locally countable weak-base.
(2)X is ag-first countable space with aσ-locally countable cs-network.
(3)X is ag-first countable space with aσ-locally countable k-network.
Proof. (1)⇒(2) is obvious.
(2) ⇒(3). Suppose X is a g-first countable space with a σ-locally countable cs-network. LetP =∪{Pn :n∈N} be aσ-locally countablecs-network for X, where eachPn is locally countable inX. We will show that P is ak-network for X. Suppose K ⊂ V with K non-empty compact and V open in X. For each n∈N, put
An={P ∈ Pn:P∩K6= Φ andP ⊂V},
thenAn is countable, and soA=∪{An:n∈N}is countable. DenoteA={Pi: i∈N}, thenK⊂ S
i≤n
Pi for some n∈N. Otherwise,K6⊂ S
i≤n
Pi for eachn∈N, so choosexn∈K\ S
i≤n
Pi. Because{P∩K:P ∈ P}is a countablecs-network for a subspace K and a compact space with a countable network is metrizable, then Kis a compact metrizable space. Thus{xn}has a convergent subsequence{xnk}, wherexnk→x. Obviouslyx∈K. SinceP is acs-network forX, then there exist m∈ N and P ∈ P such that{xnk : k≥m} ∪ {x} ⊂ P ⊂V. Now, P =Pj for somej ∈N. Takel≥msuch thatnl≥j, thenxnl ∈Pj. This is a contradiction.
Therefore, (2)⇒(3) holds.
(2) ⇒ (1). Suppose X is a g-first countable space with σ-locally countable cs-network. LetP =∪{Pm:m∈N}be a σ-locally countablecs-network forX, where eachPmis locally countable inX which is closed under finite intersections
andX ∈ Pm⊂ Pm+1, and for eachx∈X, let{B(n, x) :n∈N} be a decreasing weak neighbourhood sequence ofxinX. Put
Fm,x={P ∈ Pm:B(n, x)⊂P for some n∈N}, Fx=∪{Fm,x:m∈N}
Fm=∪{Fm,x:x∈X} F=∪{Fx:x∈X}
we will show thatF is aσ-locally countable weak-base forX.
It is easy to check thatF satisfies the condition (1), (2) of Definition 1.
SupposeGbe an open subset ofX, then for each x∈G, there existsP ∈ Fx
with P ⊂G. Otherwise, denote {P ∈ P : x∈ P ⊂G} = {P(m, x) :m ∈ N}.
Then B(n, x) 6⊂P(m, x) for each n, m∈ N, so choose xn,m ∈B(n, x)\P(m, x).
Forn≥m, letxn,m=yk, wherek=m+n(n−1)2 . The the sequence{yk:k∈N} converges to the pointx. Thus, there existm, i∈N such that{yk:k≥i} ∪ {x} ⊂ P(m, x) ⊂G because P is a cs-network for X. Take j ≥ i with yj = xn,m for somen≥m. Thenxn,m∈P(m, x). This is a contradiction. On the other hand.
If G ⊂ X satisfies that for eachx ∈ G there exists P ∈ Fx with P ⊂ G, then B(n, x)⊂Gfor some n∈N. ThusGis open inX.
HenceF is a weak-base forX.
For each m ∈ N, Fm ⊂ Pm, then Fm is locally countable in X. Thus F =
∪{Fm:m∈N} isσ-locally countable inX. Therefore, (2)⇒(1) holds.
Corollary 6. A paracompact space with a σ-locally countable weak-base is g- metrizable.
Proof. SupposeX is a paracompact space with aσ-locally countable weak-base.
By Theorem 5,X is ag-first countable space with aσ-locally countablek-network.
Since a paracompact space with aσ-locally countablek-network is anℵ-space ([9, Lemma 1]), then X is an ℵ-space. Thus X is g-metrizable by Theorem 2.4 in
[6].
In conclusion of this paper, we pose the following question in view of Theorem 5.
Question 7. Does (3)⇒(1) in Theorem 6 hold?
Acknowledgment. The author would like to thank the referee for his valuable suggestions.
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College of Mathematics and Econometrics, Hunan University Changsha, Hunan 410082, P. R. China
E-mail:[email protected]
Current address: Department of Information, Hunan Business College Changsha, Hunan 410205, P. R. China
