����������� ������ 57 (2005), 35–40

UDK 515.122.2

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A MAPPING THEOREM ON ℵ-SPACES

Zhaowen Li

Abstract. In this paper we give a mapping theorems on ℵ-spaces by means of strong compact-covering mappings and σ-mappings. As an application, we get characterizations of quotient (pseudo-open) σ-images of metric spaces.

ℵ-spaces form one class of generalized metric spaces, and play an important role in metrization theory. The concept of σ-mappings was introduced by S.Lin in [5], and by using it, σ-spaces are characterized as σ-images of metric spaces. In [17], g-metrizable spaces are characterized as weak-open σ images of metric spaces. The purpose of this paper is to establish the relationships between metric spaces and ℵ-spaces by means of strong compact-covering mappings and σ-mappings, and get characterizations of quotient (pseudo-open) σ-images of metric spaces.

In this paper all spaces are regular and T_{1}, all mappings are continuous and surjective. N denotes the set of natural numbers, ω denotes N∪{0}. For a collection P of subsets of a space X and a mapping f: X → Y, denote f(P) = {f(P) : P ∈ P}. For two families A and B of subsets of X, denote A ∧ B = {A ∩ B : A ∈ A and B ∈ B}.

Definition 1. Let P be a cover of a space X.

(1) P is a k-network for X [9] if for each compact subset K of X and its open neighborhood V, there exists a finite subcollection P′ of P such that K ⊂ ∪P′ ⊂ V.

(2) P is called a cs-network for X if for each x ∈ X, its open neighborhood V and a sequence {x_{n}} converging to x, there exists P ∈ P such that {x_{n} : n ≥ m} ∪ {x} ⊂ P ⊂ V for some m ∈ N.

(3) P is called a cs^{∗}-network for X if for each x ∈ X, its open neighborhood V and a sequence {x_{n}} converging to x, there exist P ∈ P and a subsequence {x_{n_{k}}} of {x_{n}} such that {x_{n_{k}} : k ∈ N} ∪ {x} ⊂ P ⊂ V.

A space X is called an ℵ-space if X has a σ-locally finite k-network.

AMS Subject Classification: 54 E 99; 54 C 10

Keywords and phrases: ℵ-spaces; strong compact-covering mappings; σ-mappings; k-spaces; Fréchet spaces.

Definition 2. Let f: X → Y be a mapping.

(1) f is called a σ-mapping [5] if there exists a base B for X such that f(B) is a σ-locally finite collection of subsets of Y.

(2) f is called a strong sequence-covering mapping [15] if each convergent sequence (including its limit point) of Y is the image of some convergent sequence (including its limit point) of X.

(3) f is called a sequence-covering mapping [10] if each convergent sequence (including its limit point) of Y is the image of some compact subset of X.

(4) f is a sequentially-quotient mapping [11] (resp. subsequence-covering mapping [8]) if, whenever K is a convergent sequence (including its limit) in Y, then there is a convergent sequence L (including its limit) in X (resp. compact subset L of X) such that f(L) is a subsequence of K.

(5) f is a compact-covering mapping if each compact subset of Y is the image of some compact subset of X.

(6) f is a strong compact-covering mapping [6] if it is both strong sequencecovering and compact-covering.

Theorem 3. The following are equivalent for a space X:

(1) X is an ℵ-space;

(2) X is a strong compact-covering σ-image of a metric space;

(3) X is a strong sequence-covering σ-image of a metric space;

(4) X is a sequence-covering σ-image of a metric space;

(5) X is a subsequence-covering σ-image of a metric space;

(6) X is a sequentially-quotient σ-image of a metric space;

(7) X is a compact-covering σ-image of a metric space.

Proof. (1) ⟹ (2). Suppose X is an ℵ-space. Then X has a σ-locally finite cs-network by Theorem 4 in [4]. Let P = ⋃{P_{i} : i ∈ N} be a σ-locally finite cs-network for X, where each P_{i} = {P_{α} : α ∈ A_{i}} is a locally finite collection of subsets of X which is closed under finite intersections and X ∈ P_{i} ⊂ P_{i+1}. For each i ∈ N, endow A_{i} with the discrete topology; then A_{i} is a metric space. Put

M = {α = (α_{i}) ∈ ∏_{i∈N} A_{i} : {P_{α_{i}} : i ∈ N} ⊂ P

forms a network at some point x(α) ∈ X},

A mapping theorem on ℵ-spaces 37

and endow M with the subspace topology induced from the usual product topology of the collection {A_{i} : i ∈ N} of metric spaces. Then M is a metric space. Since X is Hausdorff, x(α) is unique in X for each α ∈ M. We define f: M → X by f(α) = x(α) for each α ∈ M. Since P is a σ-locally finite cs-network for X, f is surjective. For each α = (α_{i}) ∈ M, f(α) = x(α). Suppose V is an open neighborhood of x(α) in X; there exists n ∈ N such that x(α) ∈ P_{α_{n}} ⊂ V. Set W = {c ∈ M : the n-th coordinate of c is α_{n}}; then W is an open neighborhood of α in M, and f(W) ⊂ P_{α_{n}} ⊂ V. Hence f is continuous. We will show that f is a strong compact-covering σ-mapping.

(i) f is a σ-mapping.

For each n ∈ N and α_{n} ∈ A_{n}, put

V (α_{1}, …, α_{n}) = {β ∈ M: for each i ≤ n, the i-th coordinate of β is α_{i}}. Let B = {V (α_{1}, …, α_{n}) : α_{i} ∈ A_{i}(i ≤ n) and n ∈ N}. Then B is a base for M.

To prove f is a σ-mapping, we only need to check that f(V (α_{1}, …, α_{n})) = ⋂_{i≤n} P_{α_{i}} for each n ∈ N and α_{n} ∈ A_{n} because f(B) is σ-locally finite in X by this result.

For each n ∈ N, α_{n} ∈ A_{n} and i ≤ n, f(V (α_{1}, …, α_{n})) ⊂ P_{α_{i}}, hence f(V (α_{1}, …, α_{n})) ⊂ ⋂_{i≤n} P_{α_{i}}. On the other hand, for each x ∈ ⋂_{i≤n} P_{α_{i}}, there is β = (β_{j}) ∈ M such that f(β) = x. For each j ∈ N, P_{β_{j}} ∈ P_{j} ⊂ P_{j+n}, and thus there is α_{j+n} ∈ A_{j+n} such that P_{α_{j+n}} = P_{β_{j}}. Set α = (α_{j}), then α ∈ V (α_{1}, …, α_{n}) and f(α) = x. Thus ⋂_{i≤n} P_{α_{i}} ⊂ f(V (α_{1}, …, α_{n})). Hence f(V (α_{1}, …, α_{n})) = ⋂_{i≤n} P_{α_{i}}. Therefore, f is a σ-mapping.

(ii) f is strong sequence-covering.

For each sequence {x_{n}} converging to x_{0}, we can assume that all x′_{n}s are distinct, and that x_{n} ≠ x_{0} for each n ∈ N. Set K = {x_{m} : m ∈ ω}. Suppose V is an open neighborhood of K in X. A subcollection A of P_{i} is said to have property F (K, V) if:

(a) A is finite;

(b) for each P ∈ A, ∅ ≠ P ∩ K ⊂ P ⊂ V

(c) for each z ∈ K, there exists a unique P_{z} ∈ A such that z ∈ P_{z}

(d) if x_{0} ∈ P ∈ A, then K \ P is finite.

Since P is a σ-locally finite cs-network for X, there are A ⊂ P_{i}, with F (K, V) property, and we can assume that {A ⊂P_{i} : A has the property F (K, X)} = {A_{ij} : j ∈ N}.

For each n ∈ N, put

P′_{n} = ⋀_{i,j≤n} P_{ij} ;

then P′_{n} ⊂ P_{n} and P′_{n} also has the property F (K, X).

For each i ∈ N, m ∈ ω and x_{m} ∈ K, there is α_{im} ∈ A_{i} such that x_{m} ∈ P_{α_{im}} ∈ P′_{i}. Let β_{m} = (α_{im}) ∈ ∏_{i∈N} A_{i}. It is easy to prove that {P_{α_{im}} : i ∈ N} is a network of x_{m} in X. Then there is a β_{m} ∈ M such that f(β_{m}) = x_{m} for each m ∈ ω. For each i ∈ N, there is n(i) ∈ N such that α_{in} = α_{io} when n ≥ n(i). Hence the sequence {α_{in}} converges to α_{io} in A_{i}. Thus the sequence {β_{n}} converges to β_{0} in M. This show that f is strong sequence-covering.

(iii) f is compact-covering.

Since Y has a σ-locally finite cs-network, for each compact subset L of Y, L has a countable cs-network. So L is metrizable. We can prove that f is compactcovering by the proof of Theorem 2 in [6].

(2) ⟹ (3) ⟹ (4) ⟹ (5), (6) ⟹ (5), (2) ⟹ (7) ⟹ (4) are obvious.

(5) ⟹ (6). Suppose X is the image in the metric space M under a subsequence-covering σ-mapping f. It suffices to show f is sequentially-quotient. For each convergent sequence {x_{n}} of X with x_{n} → x, set K = {x_{n} : n ∈ N}∪{x}. Since f is subsequence-covering, there exists a compact subset L of M such that f(L) is a subsequence of {x_{n}}. Denote this subsequence of {x_{n}} by {x_{n_{m}}}. For each m ∈ N, let f(a_{m}) = x_{n_{m}} with a_{m} ∈ M. Then {a_{m}} is a sequence in compact metrizable subspace L of M. Thus {a_{m}} has a convergent subsequence {a_{m_{i}}} with a_{m_{i}} → a. Obviously, f(a) = x. Put S = {a_{m_{i}} : i ∈ N} ∪{a}, then f(S) is a subsequence of K. This implies that f is sequentially-quotient.

(6) ⟹ (1). Suppose X is the image of a metric space M under a sequentiallyquotient σ-mapping f. Since f is a σ-mapping, there exists a base B for M such that f(B) is a σ-locally finite collection of subsets of X. Since sequentially-quotient mappings preserve cs^{∗}-networks by Proposition 2.7.3 in [3], f(B) is a σ-locally finite cs^{∗}-network for X. Hence X is an ℵ-space by [14, Lemma 1.17, Theorem 1.4].

Lemma 4 Suppose f is a quotient mapping from a k-space M onto a space X. If P is a k-network for M and f(P) is point-countable in X, then f(P) is a k-network for X.

Proof. Denote F = f(P). Suppose K ⊂ V with K non-empty compact and V open in X. Put

A = {F ∈ F : F ∩ K ≠ ∅ and F ⊂ V }.

Then K ⊂ ∪A′ for some finite A′ ⊂ A. Otherwise, for any finite A′ ⊂ A, K \∪A′ ≠ ∅. For each x ∈ K, put

A_{x} = {F ∈ F : x ∈ F ⊂ V };

then A_{x} is countable, and A = ∪{A_{x} : x ∈ K}. Denote A_{x} = {F_{i}(x) : i ∈ N} for each x ∈ K. Take x_{1} ∈ K; then there exists a infinite subset D = {x_{n} : n ∈ N} of K such that for each n, x_{n+1} ∈ K \ ⋃_{i,j≤n} F_{i}(x_{j}). So D has a cluster point by the compactness of K. Let x be a cluster point of D, and set B = D \ {x}; then B is not closed in X. Since f is a quotient mapping, f^{−1}(B) is not closed in M. Since M is a k-space, there exists a compact subset L of M such that f^{−1}(B) ∩ L is not closed in L. Let g = f|_{L}: L → f(L); then g is a closed mapping, and g^{−1}(B ∩ f(L)) = f^{−1}(B) ∩ L. So B ∩ f(L) is not closed in f(L). Hence B ∩ f(L) is an infinite subset of X, and D ∩ f(L) is such. By K ∩ f(L) ≠ ∅, H = L ∩ f^{−1}(K) is non-empty compact in M and H ⊂ f^{−1}(K) ⊂ f^{−1}(V), H ⊂ ∪P′ ⊂ f^{−1}(V) for some finite P′ ⊂ P. Thus f(H) ⊂ f(∪P′) ⊂ V. Denote P′ = {P_{m} : m ≤ q}. We can assume that P_{m} ∩ H ≠ ∅ for each m ≤ q; then f(P_{m}) ∈ A. Since

D ∩ f(∪P′) ⊃ D ∩ f(H) = D ∩ f(L),

D ∩ f(∪P′) is infinite. Thus f(P_{m}) includes infinitely many points of D for some m ≤ q. Take x_{j} ∈ D ∩ f(P_{m}); then f(P_{m}) = F_{i}(x_{j}) for some i ∈ N. However, there exists n > i, j such that x_{n} ∈ F_{i}(x_{j}), a contradiction. Hence K ⊂ ∪A′ ⊂ V for some finite A′ ⊂ A. So F is a k-network for X.

The following two corollaries can be proved using Theorem 3, Lemma 4 and [3, Proposition 2.1.16, Theorem 2.7.23].

Corollary 5. The following are equivalent for a space X:

(1) X is a k- and ℵ-space;

(2) X is a strong compact-covering and quotient σ-image of a metric space;

(3) X is a quotient σ-image of a metric space.

Corollary 6. The following are equivalent for a space X:

(1) X is a Fréchet space and ℵ-space;

(2) X is a pseudo-open σ-image of a metric space;

(3) X is a closed s-image of a metric space.

Example 7. Let Z be the topological sum of the unit interval [0, 1], and the collection {S(x) : x ∈ [0, 1]} of 2^{ω} convergent sequences S(x). Let X be the space obtained from Z by identifying the limit point of S(x) with x ∈ [0, 1], for each x ∈ [0, 1]. Then, from Example 2.9.27 in [3] (or Example 9.8 in [10]), we have the following facts:

(1) X is a compact-covering, quotient compact image of a locally compact metric space.

(2) X has no point-countable cs-network. From the fact above, Theorem 4 in [4], Theorem 3 and Corollary 5, the follow-

ing hold:

(a) A compact-covering compact image of a metric space need not be a compact-covering σ-image of a metric space.

(b) A quotient π-image of a metric space need not be a quotient σ-image of a metric space.

(c) A quotient s-image of a metric space need not be a quotient σ-image of a metric space. But a quotient σ-image of a metric space is a quotient s-image of a metric space.

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(received 10.06.2004)

Dept. of Math., Changsha University of Science and Technology, Changsha, Hunan 410077, P.R.China

E-mail: Lizhaowen8846@163.com