����������� ������ 50 (1998), 99–104

UDK 515.122.2

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ON NEARLY PARACOMPACT SPACES AND NEARLY FULL NORMALITY

M. N. Mukherjee and Atasi Debray

Abstract. This paper is a continuation of the study of nearly paracompact spaces, initiated by Singal and Arya in [5]. After suitably defining the generalized versions of normality and full normality in our setting, we achieve, as our final objective, an analogue of the celebrated theorem of A. H. Stone on paracompactness viz. “a Hausdorff topological spaces is paracompact iff it is fully normal”. Incidentally, in course of the deliberation, we obtain a few extended forms of certain well known results on paracompactness.

The concept of nearly paracompact spaces was initiated by Singal and Arya [5]. Although quite a good number of varied forms of paracompactness has been introduced and studied so far, the most widely studied paracompactness-like notion is near paracompactness. In addition to [5], one may refer to [1] and [4] for further descriptions concerning such a concept. In [5], Singal has furnished a comparative study of near paracompactness vis-a-vis certain other variant forms of paracompactness.

In the present article, our sole purpose is to bring forth a generalized version of the celebrated theorem of A. H. Stone that in a Hausdorff topological space paracompactness is equivalent to the full normality of the space. To this end, we have suitably defined a weaker version of fully normal space, that suits our purpose. This notion along with a corresponding introduced form of normality is developed to the extent that we need for achieving our aim. In course of the proceedings, some general forms of certain results of paracompactness are incidentally obtained.

To make the exposition self-contained, as far as practicable, we clarify certain prerequisites as follows. By a space X we shall always mean a topological space (X, τ) in which no separation axiom is presumed, and for a subset A of X, int X and cl A will respectively denote the interior and closure of A in (X, τ). By Λ we shall always mean an index set. We recall that a set A in a space X for which A = int cl A, is called a regular open set, and complements of such sets are known as regular closed sets. It is known that the class of all regular open sets in a space (X, τ) is an open base for a topology on X coarser than τ, called the semiregularization topology on X, to be denoted by τ_{s} [2]. The members of τ_{s} are known as δ-open sets of X and the complements of such sets are called δ-closed sets [7]. A cover U of a space X by open (regular open) sets is called an open cover (regular open cover) of X. For a cover U of a space X and any A ⊆ X, the set S(A, U) is defined by S(A, U) = { U ∈ U : A ∩ U ≠ ∅ }; in particular, we shall use the notation S(x, U) when A = {x}. If U and V are two covers of a space X, by U < V, we shall mean, as usual, that U is a refinement of V; we shall adopt the usual notation U^{∗} < V (U^{Δ} < V) to mean that U is a star refinement (resp. delta refinement) of V, i.e., if the covering { S(U, U) : U ∈ U } (resp. { S(x, U) : x ∈ X }) refines V. For the well known definitions of locally finite and point finite covers, and relevant results on paracompactness, one may refer to any standard text book of general topology (e.g. see Dugundji [3]).

AMS Subject Classification: 54 D 99

Keywords and phrases: Nearly paracompact spaces, nearly fully normal spaces, nearly normal spaces, almost regular spaces.

Singal and Arya [6] called a topological space X to be almost regular if for any regular closed set F and any point x ∈ X \ F, there exist disjoint open sets containing F and x respectively. The notion of near paracompactness was also initiated by the same authors in the following way:

Definition 1. [5] A topological space X is nearly paracompact if every regular open cover of X has a locally finite open refinement.

We now introduce the following definition:

Definition 2. A space X is said to be nearly fully normal if for every regular open cover U of X, there exists an open cover V of X such taht V^{∗} < U.

Remark 3. It is easy to see that a space X is nearly fully normal iff for every regular open cover U of X, there exists a regular open cover V of X such that V^{∗} < U.

Theorem 4. A space X is nearly fully normal iff every regular open cover of X has a regular open δ-refinement.

Proof. If X is nearly fully normal, then by Remark 3 and the fact that for any cover V of X, V^{Δ} < V^{∗}, the requirement is met.

Conversely, for a given regular open cover U of X, there exists a regular open cover V such that V^{Δ} < U. Now, for the regular open cover V of X, there is a regular open cover W of X such that W^{Δ} < V. So W < W^{Δ} < V and hence W^{Δ} < W^{ΔΔ} < V^{Δ}. Since W^{Δ} < W^{∗} < W^{ΔΔ}, we get the regular open cover W of X such that W^{∗} < U, proving X to be nearly fully normal.

Definition 5. A space X is said to be nearly normal if for any two disjoint sets F and G, one of which is δ-closed and the other regular closed, there exist open sets U, V such that F ⊆ U, G ⊆ V and U ∩ V = ∅.

Theorem 6. Any nearly fully normal space X is nearly normal.

Proof. Let F and G be two disjoint δ-closed sets in a nearly fully normal space X, with F a regular closed set. Then X \ F is regular open while X \ G is δ-open.

Then there exists a family { G_{α} : α ∈ Λ } of regular open sets in X such that X \ G = ⋃_{α∈Λ} G_{α}. Thus U = { X \ F, G_{α} : α ∈ Λ } is a regular open cover of X. By nearly full normality of X and Theorem 4, there exists a regular open cover B of X such that B^{Δ} < U. We consider U = S(F, B) and V = S(G, B). Then U, V are open sets in X such that F ⊆ U and G ⊆ V. In order to separate F and G strongly, we must have U ∩ V = ∅. If possible, let U ∩ V ≠ ∅. Then there exist V_{1}, V_{2} ∈ B such that V_{1} ∩ F ≠ ∅, V_{2} ∩ F ≠ ∅ and V_{1} ∩ V_{2} ≠ ∅. Hence there is p ∈ V_{1} ∩ V_{2}. Clearly, V_{1} ∪ V_{2} ⊆ S(p, B) showing that S(p, B) ∩ F ≠ ∅ and S(p, B) ∩ G ≠ ∅. Consequently, S(p, B) ⊈ X \ F and S(p, B) ⊈ X \ G = ⋃_{α∈Λ} G_{α}, i.e., S(p, B) ⊈ G_{α}, for all α ∈ Λ. This is a contradiction, since B^{Δ} < U. Hence U ∩ V = ∅, proving the space to be nearly normal.

In a Hausdorff space every singleton is clearly δ-closed. Thus in a Hausdorff nearly normal space, every regular closed set and a point outside it can be strongly separated by open sets. This leads us to infer:

Corollary 7. A Hausdorff nearly normal space is almost regular.

Corollary 8. A Hausdorff nearly fully normal space is almost regular.

Theorem 9. A nearly paracompact Hausdorff space is nearly normal.

Proof. We first show that a nearly paracompact Hausdorff space is almost regular. Let F be a regular closed set and p ∈ X \F. For each q ∈ F, by Hausdorffness of X there exist open sets V (q) and U_{q}(p) containing q and p respectively such that V (q) ∩ U_{q}(p) = ∅, so that p ∉ cl V (q). Consequently, V = { X \ F, int cl V (q) : q ∈ F } becomes a regular open cover of X. By near paracompactness of X, there exists a locally finite open cover U of X such that U < V. Now, the set S(F, U) ( = U, say) is an open set with F ⊆ U. Thus it is enough to show that p ∉ U. In fact, if U′ ∈ U is such that U′ ∩ F ≠ ∅, then U′ ⊈ X \ F. So as U < V, there exists q ∈ F such that int cl V (q) ⊇ U′. Thus cl U′ ⊆ cl V (q). As p ∉ cl V (q), p ∉ cl U′, i.e., p ∉ ⋃{ cl U′ : U′ ∈ U, U′ ∩ F ≠ ∅ } = cl[⋃{ U′ ∈ U : U′ ∩ F ≠ ∅ }] (since U is a locally finite family) = cl U.

Next, we show that X is nearly normal. Let F and G be two disjoint δ-closed sets in X with G a regular closed set. Then by almost regularity of X, to each p ∈ F there corresponds an open set V (p) such that

cl V (p) ∩ G = ∅.  (1)

As X \ F is δ-open, X \ F = ⋃_{α∈Λ} G_{α}, where { G_{α} : α ∈ Λ } is a family of regular open sets in X. Now, V = { G_{α}, int cl V (p) : α ∈ Λ, p ∈ F } is a regular open cover of X. By near paracompactness of X, there exists a locally finite open cover U of X such that U < V. We put U = S(F, U). Then

cl U = cl[⋃{ U′ ∈ U : U′ ∩ F ≠ ∅ }] = ⋃{ cl U′ : U′ ∈ U, U′ ∩ F ≠ ∅ }. (2)

Now, U′ ∩ F ≠ ∅ ⟹ U′ ⊈ X \ F = ⋃_{α∈Λ} G_{α} ⟹ U′ ⊈ G_{α}, for all α ∈ Λ. So as U < V, there exists p ∈ F such that U′ ⊆ int cl V (p). Then by (1), cl U′ ⊆ cl V (p). Thus by (2), cl U ⊆ X \ G. Consequently, F ⊆ U ⊆ cl U ⊆ X \ G, proving the space to be nearly normal.

Lemma 10. Let X be a nearly normal space. Then for every point finite regular open cover Ω = { G_{α} : α ∈ Λ } of X, there is a regular open cover { V_{α} : α ∈ Λ } of X such that cl V_{α} ⊆ G_{α}, for each α ∈ Λ; moreover, V_{α} ≠ ∅ whenever G_{α} ≠ ∅.

Proof. Let X be nearly normal. By Zermelo’s theorem, we can choose a wellorder < on Λ so that Ω = { G_{α} : α < T } where T is some definite ordinal number. We shall now define a regular open set V_{α}, for every α < T such that

X \ [⋃{ V_{β} : β < α } ∪ ⋃{ G_{γ} : γ > α }] ⊆ V_{α} ⊆ cl V_{α} ⊆ G_{α}.  (1)

For this, we apply transfinite induction on the ordinal α. We define V_{0} first. We have, X \ [⋃{ G_{γ} : γ > 0 }] is a δ-closed set in X, contained in G_{0} (as Ω is a cover of X). By near normality, there is a regular open set V_{0} such that X \ [⋃{G_{γ} : γ > 0 }] ⊆ V_{0} ⊆ cl V_{0} ⊆ G_{0}. Suppose that we have already defined V_{β}, for all β < α. Then { V_{β} : β < α } ∪ { G_{γ} : γ ⩾ α } forms a covering of X. In fact, if p ∈ X is such that p ∉ G_{γ}, for all γ ⩾ α, by point finiteness of Ω there exists the last ordinal β < α such that p ∈ G_{β}. If p ∉ V_{β′} for all β′ < β we have by (1), p ∈ X \ [⋃{ V_{β′} : β′ < β } ∪ ⋃{ G_{β′′} : β′′ > β }] ⊆ V_{β}, so that p ∈ V_{β}. Now,

X \ [⋃{ V_{β} : β < α } ∪ ⋃{ G_{γ} : γ > α }] ⊆ G_{α}.

By near normality of X, there exists a regular open set V_{α} such that

X \ [⋃{ V_{β} : β < α } ∪ ⋃{ G_{γ} : γ > α }] ⊆ V_{α} ⊆ cl V_{α} ⊆ G_{α}.

Thus we can construct V_{α} satisfying (1), for each α < T, i.e., V_{α} is a regular open set satisfying cl V_{α} ⊆ G_{α}, for each α < T. Finally, we are to show that { V_{α} : α < T } covers X. Choose p ∈ X. The covering Ω of X being point finite, there is the last ordinal number β < T such that p ∈ G_{β}. Then p ∉ ⋃{ G_{γ} : γ > β }. Now, if p ∉ V_{β′}, for each β′ < β, we have p ∈ X \ [[⋃{ G_{γ} : γ > β }] ∪ [⋃{ V_{β′} : β′ < β }]] ⊆ V_{β}. Thus p ∈ V_{β}. Consequently, we get a regular open cover { V_{α} : α ∈ Λ } with cl V_{α} ⊆ G_{α}, for each α ∈ Λ.

Lemma 11. Every regular open locally finite covering of a nearly normal space has a regular open delta refinement.

Proof. Suppose V = { V_{α} : α ∈ Λ } is a locally finite regular open cover of a nearly normal space X. We construct, by Lemma 10, a regular open cover W = { W_{α} : α ∈ Λ } such that cl W_{α} ⊆ V_{α}, for all α ∈ Λ. Let Λ′ be a subset of the index set Λ. We define

P (Λ′) = [⋂{ V_{α} : α ∈ Λ′ }] ∩ [⋂{ X \ cl W_{α} : α ∈ Λ \ Λ′ }].

Let us prove that P (Λ′) is a regular open set. Since V is locally finite, \overline{W} ( = { cl W : W ∈ W }) is also so. Then

⋂{ X \ cl W_{α} : α ∈ Λ \ Λ′ } = X \ ⋃{ cl W_{α} : α ∈ Λ \ Λ′ }
= X \ cl[⋃{ W_{α} : α ∈ Λ \ Λ′ }].

On the other hand, as V is locally finite, ⋂{ V_{α} : α ∈ Λ′ } = ∅, if Λ′ is infinite, and is regular open if Λ′ is finite. Thus P (Λ′) is a regular open set, for each subset Λ′ of Λ. For any p ∈ X, p ∈ P (Λ′) whenever Λ′ = { α ∈ Λ : p ∈ cl W_{α} }. Thus P = { P (Λ′) : Λ′ ⊆ Λ } is a regular open cover of X. To prove P^{Δ} < V, we consider a given point p of X. Since W is a covering of X, p ∈ W_{β} for some β ∈ Λ. Let p ∈ P (Λ′), then in view of the definition of P (Λ′), we know that β ∈ Λ′; because if β ∈ Λ \ Λ′, then p ∉ X \ cl W_{β} ⊇ P (Λ′), a contradiction as p ∈ P (Λ′). Thus P (Λ′) ⊆ V_{β}. As the inclusion holds for all P (Λ′) which contains p, S(p, P) ⊆ V_{β}, proving that P^{Δ} < V.

As a last step towards our prerequisites for attaining our desired goal we recall the following result from [5].

Lemma 12. An almost regular space X is nearly paracompact iff each regular open cover of X has an open σ-locally finite refinement.

With the deliberations so far, we are now set to prove the desired analogue of the celebrated Stone’s theorem as follows.

Theorem 13. A Hausdorff space X is nearly paracompact iff it is nearly fully normal.

Proof. Let us first assume that X is Hausdorff and nearly paracompact. Then by Theorem 9, X is nearly normal. If U is any regular open cover of X, then U has a regular open locally finite refinement V. By Lemma 11, there exists a regular open delta refinement W of V, i.e., W^{Δ} < V. Then W^{Δ} < U. Then by Theorem 4, X is nearly fully normal.

Conversely, let X be nearly fully normal Hausdorff space. Let Ω = { G_{α} : α ∈ I } (I being an index set) be a regular open cover of X. We define a sequence of regular open covers {Ω_{i}} of X taking Ω_{0} = Ω and Ω_{n+1} to be a regular open star refinement of Ω_{n} for n = 0, 1, 2,…. Let G_{α,n} = { x ∈ X : there is a regular open set V containing x such that S(V, Ω_{n}) ⊆ G_{α} }, for every α ∈ I and n = 1, 2,…. We show that for each n, the family {G_{α,n} : α ∈ I } is an open cover of X, which is a refinement of Ω. We apply induction on n to prove that { G_{α,n} : α ∈ I } is a covering of X for each n ∈ N (N denoting the set of all natural numbers). Let x ∈ X. Then for some V^{1} ∈ Ω_{1}, x ∈ V^{1}. Since Ω_{1}^{∗} < Ω, there is G_{α} ∈ Ω such that S(V^{1}, Ω_{1}) ⊆ G_{α}. Hence by definition of G_{α,1} we have x ∈ G_{α,1}. Thus { G_{α,1} : α ∈ I } is a cover of X.

Suppose that { G_{α,n−1} : α ∈ I } is a cover of X. Then for any x ∈ X, there is a regular open set V containing x such that S(V, Ω_{n−1}) ⊆ G_{α}, for some α ∈ I, i.e., x ∈ G_{α,n−1}. But Ω is a cover of X and hence x ∈ V^{n}, for some V^{n} ∈ Ω. As Ω_{n}^{∗} < Ω_{n−1}, there is T^{n−1} ∈ Ω_{n−1} such that S(V^{n}, Ω_{n}) ⊆ T^{n−1}. Since x ∈ S(V^{n}, Ω_{n}), we have x ∈ T^{n−1} ∈ Ω_{n−1}. Hence x ∈ V ∩ T^{n−1}, i.e., V ∩ T^{n−1} ≠ ∅. Thus T^{n−1} ⊆ S(V, Ω_{n−1}) ⊆ G_{α}, so that S(V^{n}, Ω_{n}) ⊆ T^{n−1} ⊆ G_{α}. Thus x ∈ G_{α,n} and hence { G_{α,n} : α ∈ I } is a cover of X, for each n ∈ N. We prove that if x ∈ G_{α,n} and y ∉ G_{α,n+1}, then there does not exist any G ∈ Ω_{n+1} such that x, y ∈ G. (1)

For every G ∈ Ω_{n+1}, there exists H ∈ Ω_{n}, such that S(G, Ω_{n+1}) ⊆ H (since Ω_{n+1}^{∗} < Ω_{n}). Thus x ∈ G ∩ G_{α,n} ⟹ H ⊆ S(x, Ω_{n}) ⊆ G_{α}. In fact, as x ∈ G, x ∈ S(G, Ω_{n+1}) and hence x ∈ H, where H ∈ Ω_{n} and hence H ⊆ S(x, Ω_{n}). Again x ∈ G_{α,n} implies that there exists some open neighbourhood V of x such that S(V, Ω_{n}) ⊆ G_{α}. As x ∈ V, all those members of Ω_{n} which contain x have nonempty intersection with V and hence each is contained in S(V, Ω_{n}); then S(x, Ω_{n}) ⊆ G_{α}. Now, as S(G, Ω_{n+1}) ⊆ H, for some H ∈ Ω_{n}, and H ⊆ G_{α}, we obtain S(G, Ω_{n+1}) ⊆ H. Now for z ∈ G, as G is a regular open set containing x with S(G, Ω_{n+1}) ⊆ G_{α}, we have z ∈ G_{α,n+1} so that G ⊆ G_{α,n+1}. Thus y ∉ G.

Let us now suppose that the set I is well-ordered by the relation ‘<’ and let

H_{β,n} = G_{β,n} \ cl[ ⋃_{α<β} G_{α,n+1}], for every β ∈ I and each n ∈ N.

For any pair of distinct elements γ, η ∈ I, we have either γ < η or γ > η and correspondingly we have H_{η,n} ⊆ X \ G_{γ,n+1} or H_{γ,n} ⊆ X \ G_{η,n+1}. Now, if x ∈ H_{γ,n} and y ∈ H_{η,n} where γ ≠ η (suppose γ < η), there is no set G ∈ Ω_{n+1} which would contain both x and y. Indeed, x ∈ H_{γ,n} means x ∈ G_{γ,n}. Again y ∈ H_{η,n} ⊆ X \ G_{γ,n+1} means that y ∉ G_{γ,n+1}. Then by (1), there is no G ∈ Ω_{n+1} which contains both x and y. Thus we see that for every x ∈ X, there is some G ∈ Ω_{n+1} such that x ∈ G and G intersects at most one H_{α,n}, for a fixed n ∈ N (H_{α,n}’s being mutually disjoint). Hence { H_{α,n} : α ∈ I } is a discrete family of open sets, for n = 1, 2,….

Finally we shall show that { H_{α,n} : α ∈ I, n ∈ N } is a cover of X. Let y ∈ X, and let α_{n} is the first index such that y ∈ G_{α_{n},n} (that { G_{α,n} : α ∈ I } is a cover of X, for each n ∈ N, has already been proved). Taking inf{ α_{n} : n ∈ N } = α(y) (say), we obtain y ∈ G_{α(y),n}, for some n ∈ N. So for α < α(y) ⩽ α_{n+2}, y ∉ G_{α,n+2}. For α < α(y), let x ∈ G_{α,n+1}. Then there does not exist any G ∈ Ω_{n+2} such that x and y both belong to G. So the collection of sets in Ω_{n+2} which contain y cannot intersect ⋃{ G_{α,n+1} : α < α(y) }. Now, H_{α(y),n} = G_{α(y),n} \ cl[⋃_{α<α(y)} G_{α,n+1}] and y ∈ G_{α(y),n}. But since S(y, Ω_{n+2}) ∩ [⋃{ G_{α,n+1} : α < α(y) }] = ∅, y ∉ cl[⋃_{α<α(y)} G_{α,n+1}]. Thus y ∈ H_{α(y),n}. Hence every regular open cover of X has an open σ-discrete refinement. Every σ-discrete refinement being σ-locally finite, it follows by virtue of Corollary 8 and Lemma 12 that X is nearly paracompact.

REFERENCES

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(received 11.11.1997.)

Department of Pure Mathematics, University of Calcutta, 35, Ballygunge Circular Road, Calcutta–700 019, India