����������� ������ 52 (2000), 119–123

UDK 515.122.2

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A NOTE ON SUBPARACOMPACT SPACES

D. Buhagiar and S. Lin

Abstract. Some results on subparacompact spaces have only been obtained for the class of regular spaces. The aim of this note is to obtain some properties of subparacompact spaces, and also some applications to generalized metric spaces, without requiring the regularity of the spaces involved.

Subparacompactness is one of the important covering properties in General Topology. D. K. Burke [2] and H. J. K. Junnila [7] have made many contributions to the characterizations and applications of subparacompact spaces. Some results on subparacompact spaces have only been obtained for the class of regular spaces. In particular, we have the following two results:

(1) A regular inverse image of a subparacompact space under a perfect map is a subparacompact space [2].

(2) Every regular, strong Σ-space is a subparacompact space [4,9].

In [1], D. Buhagiar and T. Miwa investigated four proerties on continuous maps without requiring the spaces to satisfy any separation axioms. In particular, they proved that the inverse image of a subparacompact space under a subparacompact map is a subparacompact space. In this paper we use a technique from[1] about subparacompact maps obtaining some further properties of subparacompact spaces, and also some applications to generalized metric spaces, without requiring the regularity of the spaces involved.

In this paper all spaces are Hausdorff (T_{2}) and all maps are continuous. For a space X, if U and V are families of subsets of X, and G is a subset of X, then we denote the family { U ∩ V : U ∈ U, V ∈ V } by U ∧ V and the family { U ∩ G : U ∈ U } by U_{G}. If X is a space and A ⊂ X then by cl_{X} A we understand the closure of A in X.

AMS Subject Classification: 54 D 20, 54 E 20, 54 E 18, 54 C 10

Keywords and phrases: Subparacompact space, semi-stratifiable space, (strong) Σ- [Σ^{∗}-] space, isocompact space.

The research of the second author was supported by the NSF of Fujian Province under grant number A97025.

Lemma 1. Let K be a compact subset of a space X. If U is an open in X cover of K, then there exists an open subset O in X such that K ⊂ O and U_{O} has a finite refinement closed in O.

Proof. Since U is an open in X cover of K, there exists a finite subfamily { U_{i} : i ⩽ n } of U such that K ⊂ ⋃{ U_{i} : i ⩽ n }. For every x ∈ K, there exists i(x) ⩽ n satisfying x ∈ U_{i(x)}. Let F (x) = K \ U_{i(x)}. Since the space X is T_{2} and F (x) is a compact subset of X, there exist open subsets V (x) and W (x) in X such that x ∈ V (x), F (x) ⊂ W (x) and V (x) ∩ W (x) = ∅. Let G(x) = U_{i(x)} ∪ W (x), then G(x) is open in X and K ⊂ G(x). Take a finite subset { x_{j} : j ⩽ m } of K satisfying K ⊂ ⋃{ V (x_{j}) : j ⩽ m } and let O = (⋂_{j⩽m} G(x_{j})) ∩ (⋃_{j⩽m} V (x_{j})) ∩

(⋃_{i⩽n} U_{i}). Then, the set O is open in X and contains K. For every j ⩽ m, let P_{j} = cl_{O}(V (x_{j}) ∩ O), then we have that P_{j} is closed in O, O = ⋃_{j⩽m} P_{j}, and

P_{j} \ U_{i(x_{j})} ⊂ (G(x_{j}) \ U_{i(x_{j})}) ∩ cl_{X}(V (x_{j})) ⊂ W (x_{j}) ∩ cl_{X}(V (x_{j})) = ∅.

Consequently, P_{j} ⊂ U(i(x_{j})) and { P_{j} : j ⩽ m } is a closed refinement of U_{O} in O.

Recall that a space X is said to be subparacompact if every open cover of X has a σ-discrete closed refinement. This condiditon was introduced by L. F. McAuley [8] as F_{σ}-screenable. A map f: X → Y is said to be perfect if it is closed, onto and f^{−1}(y) is compact for every y ∈ Y.

Theorem 2. Let f: X → Y be a perfect map. If Y is a subparacompact space, then X is also a subparacompact space.

Proof. Let U be an open cover of X. For every y ∈ Y, f^{−1}(y) is compact in X and therefore, by Lemma 1, there exists an open subset O(y) in X such that f^{−1}(y) ⊂ O(y) and U_{O(y)} has a finite refinement closed in O(y). Since the map f is closed, there exists an open subset H(y) in Y such that y ∈ H(y) and f^{−1}(H(y)) ⊂ O(y). Thus, U_{f^{−1}(H(y))} has a finite closed refinement F_{y} in f^{−1}(H(y)). Let H = { H(y) : y ∈ Y }. Since the space Y is subparacompact, H has a σ-discrete closed refinement W = ⋃_{i∈N} W_{i}, here each W_{i} = { W_{i,y} : y ∈ Y } is discrete and closed in Y, and W_{i,y} ⊂ H(y). For every i ∈ N, let

P_{i} = { f^{−1}(W_{i,y}) ∩ F : y ∈ Y and F ∈ F_{y} }.

To complete the proof of the theorem, it suffices to show that P = ⋃_{i∈N} P_{i} is a σ-discrete closed refinement of U in X. The fact that P is a refinement of U is evident. Since f^{−1}(W_{i}) is discrete in X and F_{y} is finite for every y ∈ Y, we have that P_{i} is σ-discrete in X for every i ∈ N. For every y ∈ Y and every F ∈ F_{y}, let

L_{x} = \{ \begin{matrix}X \ f^{−1}(W_{i,y}), & if x ∈ X \ f^{−1}(W_{i,y}); \\ f^{−1}(H(y)) \ F, & if x ∈ f^{−1}(W_{i,y}) \ F , \end{matrix}

whenever x ∈ X \ (f^{−1}(W_{i,y}) ∩ F). Then L_{x} is an open neighbourhood of x in X and L_{x} ∩ (f^{−1}(W_{i,y}) ∩ F) = ∅. This shows that f^{−1}(W_{i,y}) ∩ F is closed in X and consequently, X is a subparacompact space.

Semi-stratifiable spaces were introduced and studied by G. D. Creede [3]. Recall that a space X is semi-stratifiable if there exists a function G which assigns to each n ∈ N and closed set H ⊂ X, an open set G(n, H) containing H and satisfying (i) H = ⋂_{n∈N} G(n, H); (ii) H ⊂ K ⟹ G(n, H) ⊂ G(n, K). As an application of Theorem2, we obtain a pull-back theorem for semi-stratifiable spaces.

Corollary 3. Let f: X → Y be a perfect map. If the space X has a G_{δ}- diagonal, and Y is a semi-stratifiable space, then X is also semi-stratifiable.

Proof. Since the space Y is semi-stratifiable, it is a subparacompact space [4, Theorem 5.11]. Hence, by Theorem 2, X is also a subparacompact space. Since X is a subparacompact space with a G_{δ}-diagonal, X is a σ^{♯}-space (i.e., a space with a σ-closure preserving closed p-network) [5, Proposition 4.1]. Since a semistratifiable space is a β-space (i.e. a space (X, τ) having a function g: N × X → τ satisfying (i) x ∈ g(n, x), and (ii) if x ∈ g(n, x_{n}), then the set {x_{n} : n ∈ N} has a cluster point in X [6]) and an inverse image of a β-space under a perfect map is a β-space, the space X is a σ^{♯}-space and a β-space. Consequently, X is a semi-stratifiable space [5, Theorem 5.2].

Remember that a space X is said to be a (strong) Σ-space [Σ^{∗}-space] if there is a σ-locally finite [σ-hereditarily closure-preserving] family P of closed subsets of X, and a cover C of X consisting of closed countably compact (compact) subsets, such that, whenever C ∈ C and C ⊂ U for some open subset U of X, then C ⊂ P ⊂ U for some P ∈ P. The family P is called a (mod k)-network (with respect to C). It is convenient to consider the following modification of (mod k)-networks. A family P of subsets of X is called an almost (mod k)-network [9] for X if every x ∈ X is in some compact C(x) ⊂ X such that, whenever C(x) ⊂ U for some open subset U of X, then x ∈ P ⊂ U for some P ∈ P.

Lemma 4. Every strong Σ^{∗}-space is a subparacompact space.

Proof. Let X be a strong Σ^{∗}-space, and let P be a σ-hereditarily closurepreserving closed (mod k)-network with respect to a cover C of compact subsets of X. If U is an open cover of X, for every x ∈ X, there exists a compact subset C(x) ∈ C with x ∈ C(x). By Lemma 1, there is an open subset O(x) in X satisfying C(x) ⊂ O(x) and U_{O(x)} has a finite closed refinement F(x) in O(x). Then, C(x) ⊂ P (x) ⊂ O(x) for some P (x) ∈ P. Let P′ = { P (x) : x ∈ X }, and for each P ∈ P′ choose a point x_{P} such that P (x_{P}) = P. Then, since P is σ-hereditarily closure-preserving, the family F = { P ∩ F : P ∈ P′ and F ∈ F(x_{P}) } is a σclosure-preserving closed refinement of U in X, and hence, X is a subparacompact space.

The proof of the following lemma can be fonud in [9].

Lemma 5. Let P be a point-countable family of closed subsets of a space X which is closed under finite intersections. If P is an almost (mod k)-network of X, there is a cover C of compact subsets of X such that P is a (mod k)-network with respect to C.

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

(1) X is a strong Σ-space;

(2) X is a subparacompact space;

(3) X is an isocompact Σ-space;

(4) X has a σ-discrete closed (mod k)-network with respect to a cover of compact subsets.

Proof. We only need to show the implication (1) ⟹ (4). Let ⋃_{i∈N} P_{i} be a σlocally finite closed (mod k)-network with respect to a cover C of compact subsets of X, here each P_{i} is locally finite in X. Since X is a subparacompact space, for every i ∈ N, tehre is a σ-discrete closed cover F_{i} of X such that each element of F_{i} intersects only finitely many elements of P_{i}. Thus, P_{i} ∧ F_{i} is σ-discrete and closed in X. Let P = { ⋂ F : F is a finite subfamily of ⋃_{i∈N}(P_{i} ∧ F_{i}) }. Then P is a σ-discrete family of closed subsets of X which is closed under finite intersections. For every x ∈ C ∈ C and every open subset U ⊃ C of X, there exist i ∈ N, P ∈ P_{i} and F ∈ F_{i} such that x ∈ F and C ⊂ P ⊂ U. Therefore, P ∩ F ∈ P and x ∈ P ∩ F ⊂ U, which shows that P is an almost (mod k)-network of X. By Lemma 5, P is a σ-discrete closed (mod k)-network with respect to a cover of compact subsets of X.

Remark 1. H. J. Junilla [7] proved the equivalence (1) ⟺ (4) in Theorem 6 for the class of regular spaces.

Theorem 7. Let f: X → Y be a closed map with Lindelöf fibres. If X is a strong Σ-space, then so is Y.

Proof. Since X is a strong Σ-space, by Lemma 5 we have that X is subparacompact. Therefore, since f is closed, Y is also subparacompact [2]. Suppose that ⋃_{i∈N} P_{i} is a σ-discrete closed (mod k)-network with respect to a cover K of compact subsets of X. For every i ∈ N, tehre is an open cover U_{i} of X such that each element of U_{i} intersects at most one element of P_{i}. Since f^{−1}(y) is Lindelöf, for every y ∈ Y, there exists a countable subfamily U_{i}(y) of U_{i} such that f^{−1}(y) ⊂ ⋃ U_{i}(y). Thus, f^{−1}(V_{i}(y)) ⊂ ⋃ U_{i}(y) for some neighbouhood V (y) of y in Y. Let V_{i} = { V_{i}(y) : y ∈ Y }. Since V_{i} is an open cover of Y, V_{i} has a σ-discrete closed refinement F_{i} = { F_{α} : α ∈ A_{i} }. Now, for each α ∈ A_{i}, there exists y_{α} ∈ Y satisfying F_{α} ⊂ V_{i}(y_{α}). Let U_{i}(y_{α}) = { U_{α}(j) : j ∈ N } and let B_{ij} = { f^{−1}(F_{α}) ∩ U_{α}(j) : α ∈ A_{i} }. Then B_{i} = ⋃_{j∈N} B_{ij} is a cover of X. Also, let C_{i} = (P_{i}∧B_{i})^{−} = { cl_{X}(P ∩B) : P ∈ P_{i}, B ∈ B_{i} }. Since P_{i} is closed and discrete in X, each of its elements is a union of some subfamily of C_{i}. Let P_{i} = { P_{γ} : γ ∈ Γ_{i} }, then C_{i} = ⋃_{j∈N} C_{ij}, where C_{ij} = { cl_{X}(P_{γ} ∩ f^{−1}(F_{α}) ∩ U_{α}(j)) : α ∈ A_{i}, γ ∈ Γ_{i} }, so that f(C_{ij}) = { cl_{X}(f(P_{γ} ∩ U_{α}(j)) ∩ F_{α}) : α ∈ A_{i}, γ ∈ Γ_{i} }. Since each U_{α}(j) intersects at most one element of P_{i} and F_{i} is σ-discrete and closed in X, we have that f(C_{ij}) is σ-discrete and closed in Y. To complete the proof, by Lemma 5, it is sufficient to show that ⋃_{i,j∈N} f(C_{ij}) is an almost (mod k)-network for Y.

For every y ∈ Y, there exists x ∈ K for some K ∈ K, satisfying y = f(x). For an open subset W ⊃ f(K) in Y, there exists γ ∈ Γ_{i} with K ⊂ P_{γ} ⊂ f^{−1}(W). Since P_{γ} is a union of some subfamily of C_{i}, there exist α ∈ A_{i} and j ∈ N such that

x ∈ cl_{X}(P_{γ} ∩ f^{−1}(F_{α}) ∩ U_{α}(j)) ⊂ P_{γ} ⊂ f^{−1}(W),

so that

y ∈ cl_{X}(f(P_{γ} ∩ U_{α}(j)) ∩ F_{α}) ⊂ f(P_{γ}) ⊂ W.

Consequently, ⋃_{i,j∈N} f(C_{ij}) is a σ-discrete closed, almost (mod k)-network for Y, and therefore, Y is a strong Σ-space.

Remark 2. E. Michael [9] proved that strong Σ-spaces are preserved by σlocally finite maps for the class of regular spaces. This is not true for the class of Hausdorff spaces. Indeed, let Y be the space with a point irrational extension topology ([10], Example 69). Then Y is a Hausdorff, second countable space which is not subparacompact. Since Y is second countable, there exists a separable metric space X and a map f: X → Y. Then, X is a strong Σ-space and f is a σ-locally finite map, while Y is not a strong Σ-space.

REFERENCES

[1] D. Buhagiar and T. Miwa, Covering properties on maps, Q & A in General Toplogy 16, 1 (1998), 53–66.

[2] D. K. Burke, On subparacompact spaces, Proc. Amer. Math. Soc. 23 (1969), 655–663.

[3] G. D. Creede, Concerning semi-stratifiable spaces, Pacific J. Math. 32 (1970), 47–54.

[4] G. Gruenhage, Generalized metric spaces, Handbook of Set-Theoretic Topology, North-Holland, 1984, pp. 423–501.

[5] R. E. Hodel, Moore spaces and ωδ-spaces, Pacific J. Math. 38 (1971), 641–652.

[6] R. E. Hodel, Spaces defined by sequences of open covers which guarantee that certain sequences have cluster points, Duke Math. J. 39 (1972), 253–263.

[7] H. J. K. Junilla, Three covering properties, Surveys in General Toplogy, Academic Press, 1980, pp. 195–245.

[8] L. F. McAuley, A note on complete colectionwise normality and paracompactness, Proc. Amer. Math. Soc. 9 (1958), 796–799.

[9] E. Michael, σ-locally finite maps, Proc. Amer. Math. Soc. 65 (1977), 159–164.

[10] L. A. Steen and J. A. Seebach, Jr., Counterexamples in Topology, Springer-Verlag, New York, 1978.

(received 22.05.2000.)

Department of Mathematics, Faculty of Science, University of Malta, Msida MSD.06, Malta E-mail: dbuha@maths.um.edu.mt Department of Mathematics, Ningde Teachers’ College, Ningde, Fujian 352100, P.R.China