����������� ������ 50 (1998), 23–29

UDK 515.122.2

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ON NEARLY PARACOMPACT SPACES VIA REGULAR EVEN COVERS

M. N. Mukherjee and Atasi Debray

Abstract. The concept of near paracompactness is a well known weaker form of paracompactness, first introduced by Singal and Arya [6], followed by its further pursuit by many others. In the present paper, the same concept has been investigated in terms of a certain type of cover, called regular even cover. A number of characterizations of almost regular nearly paracompact spaces, including those analogous to the famous Michael’s results on a regular paracompact space, have been achieved.

Paracompactness is one of the most celebrated concepts of general topology, and many of its variant forms have so far been investigated. Of the variegated generalizations of paracompactness found in the literature, mention may be made of near paracompactness, almost paracompactness, mildly paracompact spaces, mand mn-paracompactness etc. A somewhat detailed survey of some weaker and stronger forms of paracompatness may be found in [5]. In [6], Singal and Arya introduced and studied nearly paracompact spaces. The behaviour of such a space in relation to near compactness has been seen to be similar to the role of paracompactness vis-à-vis compactness. Near paracompactness has further been taken up for study in [1] and [4]. All these investigations have furnished the generalized versions of many of the well known results on paracompactness.

This paper is a continuation of the study of nearly paracompact spaces in terms of regular even covers, a concept introduced here. We ultimately achieve a number of characterizations of such a space in an almost regular space, some of which present the analogues of the famous Michael’s theorem on paracompact spaces in the presence of the axiom of regularity.

In what follows, by a space X we shall mean a topological space (X, τ). For a subset A of X, int A and cl A will stand respectively for interior and closure of A in (X, τ). The diagonal Δ_{X} of X is, as usual, given by Δ_{X} = {(x, x) : x ∈ X}. For two covers U and V of X, we shall write U < V to mean that U is a refinement of V, where we assume that a refinement of a cover is necessarily a cover of X. If V is any subset of X × X and A ⊆ X, then by V [A] we shall denote the set ⋃{V [x] : x ∈ A}, where V [x] = {y ∈ X : (x, y) ∈ V }. We recall that a set A in a space X is called regular open if A = int cl A, and complements of regular open sets are called regular closed. The class of all regular open sets in a space (X, τ) is a base for a topology on X coarser than τ, called the semiregularization topology on X and is denoted by τ_{s} [2]. The members of τ_{s} are called δ-open sets of X [8] and the complements of such sets are known as δ-closed sets [8]. The following result, to be used in the sequel, can be proved in a straightforward manner (see [3] for details).

AMS Subject Classification: 54 D 20

Keywords and phrases: Nearly paracompact spaces, almost regular spaces, regular even cover, strongly locally finite cover, strongly discrete cover.

Theorem 1. For a topological space (X, τ), (τ × τ)_{s} = τ_{s} × τ_{s}.

A family U of subsets in a space X will be called strongly locally finite (strongly discrete) if for each x ∈ X there exists a regular open set V containing x such that V intersects at most finitely members (at most one member) of U. A cover U of X is said to have a σ-strongly locally finite (σ-strongly discrete) refinement V if V = ⋃_{n=1}^{∞} V_{n}, where each V_{n} is a strongly locally finite (resp. strongly discrete) family and V < U. A set A in a space X is called a δ-neighbourhood (henceforth abreviated as nbd.) of a point x of X if there is a δ-open set U such that x ∈ U ⊆ A. If, in addition, A itself is δ-open, we shall term A as a δ-open nbd. of x. According to Singal and Arya [6], a topological space X is called nearly paracompact if every regular open cover of X has a locally finite open refinement. Let us now set the following definition:

Definiton 2. A cover U of a topological space X is said to be regular even, if there exists some δ-nbd. V of the diagonal Δ_{X} in the product space X × X such that {V [x] : x ∈ X} < U.

Theorem 3. Let U be a regular open cover of a topological space X, which has a regular closed locally finite (strongly locally finite) refinement R. Then U is a regular even cover.

Proof. For each A ∈ R, we can choose U_{A} ∈ U such that A ⊆ U_{A}. Define V_{A} = (U_{A} × U_{A}) ∪ ((X \ A) × (X \ A)), for each A ∈ R. Then each V_{A} is clearly a δ-open nbd. of the diagonal Δ_{X} in X × X. Now, for each x ∈ A, we first show that V_{A}[x] = U_{A}. In fact, x ∈ A ⟹ x ∉ (X \ A). So, for all y ∈ X,

(x, y) ∉ (X \ A) × (X \ A).  (1)

Now, y ∈ V_{A}[x] ⟹ (x, y) ∈ V_{A} ⟹ (x, y) ∈ U_{A} × U_{A} (by (1)) ⟹ y ∈ U_{A}. Again, y ∈ U_{A} ⟹ (x, y) ∈ U_{A} × U_{A} (since A ⊆ U_{A}) ⟹ (x, y) ∈ V_{A} ⟹ y ∈ V_{A}[x]. Hence V_{A}[x] = U_{A}, for each x ∈ A. Put V = ⋂{V_{A} : A ∈ R}. Then for each x ∈ X, there exists A ∈ R such that x ∈ A and then V [x] ⊆ V_{A}[x] = U_{A}. Consequently, {V [x] : x ∈ X} < U.

Now, we are left to show that V is a δ-nbd. of Δ_{X} in X ×X. Let us choose any (x, x) ∈ Δ_{X} (x ∈ X). By local finitness (strong local finitness) of R, there exists an open (resp. a regular open) set W_{x} containing x in X such that W_{x} ∩ A ≠ ∅ for at most finitely many A ∈ R, say for A_{1}, A_{2}, …, A_{k}. Our claim is:

(int cl W_{x} × int cl W_{x}) ∩ V_{A_{1}} ∩ V_{A_{2}} ∩ ⋯ ∩ V_{A_{k}} ⊆ V.  (2)

Indeed, for any A ∈ R \ {A_{1}, A_{2},…, A_{k}}, we have: W_{x} ∩ A = ∅ ⟹ W_{x} ⊆ X \ A ⟹ int cl W_{x} ⊆ int cl(X \ A) = X \ A. Thus (int cl W_{x} × int cl W_{x}) ⊆ (X \ A) × (X \ A) ⊆ V_{A}, for all A ≠ A_{1}, A_{2}, …, A_{k}, i.e. (int cl W_{x} × int cl W_{x}) ⊆ {V_{A} : A ∈ R, A ≠ A_{1}, A_{2},…, A_{k}}, from which (2) follows. Now, the left side of (2) is a δ-open set containing (x, x) so that the arbitrariness of (x, x) ensures that V is a δ-nbd. of Δ_{X}.

Definition 4. [7] A topological space X is said to be almost regular if for any regular closed set F and any point x ∈ X \ F, there exists disjoint open sets containing F and x respectively.

The following theorem was obtained by Singal and Arya in [6]:

Theorem 5. In an almost regular space X the following hold:

(a) If a regular open cover of X has a locally finite refinement then it has a closed locally finite refinement.

(b) If a regular open cover of X has a locally finite closed refinement then it has a locally finite regular closed refinement.

Remark 6. The above theorem remains equally valid (without much changes in the proof) if each of the locally finite refinements involved in the statements is a strongly locally finite refinement.

Theorem 7. Let X be a topological space such that each regular open cover of X is regular even. If U is a δ-nbd. of the diagonal Δ_{X} in X × X, then there is a symmetric δ-open nbd. V of the diagonal Δ_{X} such that V ^{∘} V ⊆ U, where V ^{∘} V = {(x, y) ∈ X × X : (x, z), (z, y) ∈ V, for some z ∈ X}.

Proof. As U is a δ-nbd of Δ_{X}, we have (using Theorem 1) that for each x ∈ X, there is a regular open nbd. W_{x} of x with W_{x} × W_{x} ⊆ U. Since W = {W_{x} : x ∈ X} is a regular open cover of X, it is a regular even cover of X and consequently, there is some δ-nbd. V_{1} of the diagonal Δ_{X} in X × X such that {V_{1}[x] : x ∈ X} < W and hence V_{1}[x] × V_{1}[x] ⊆ U. Obviously, we can regard V_{1} to be δ-open. Let V = V_{1} ∩ V_{1}^{−1}. Clearly V is symmetric and contains Δ_{X}. Now we see that if f_{x} (x ∈ X) and g are the maps given by f_{x}: X → X × X, g: X × X → X × X such that f_{x}(y) = (y, x) and g(x, y) = (y, x), where X and X × X are equipped with the topologies τ_{s} and τ_{s} × τ_{s} respectively, then f_{x} ^{∘} p_{2} = g, p_{2} : X × X → X being the second projection map. Now, p_{2} and f_{x} being continuous, so is g. Thus for a δ-open set U of X (i.e. U ∈ τ_{s}), g^{−1}(U) (= U^{−1}) is also δ-open. Hence V_{1} being a δ-open set, so is V_{1}^{−1} and hence so is V_{1} ∩ V_{1}^{−1}, i.e., V. Thus V is a symmetric δ-open nbd. of Δ_{X} such that V [x] × V [x] ⊆ U, for all x ∈ X. Since V ^{∘} V = ⋃_{x∈X}(V [x] × V [x]) (V being symmetric) it follows that V ^{∘} V ⊆ U.

Theorem 8 Let X be a topological space such that each regular open cover is regular even and let R be a strongly locally finite (strongly discrete) family of subsets of X. Then there is a δ-open nbd. W of the diagonal in X × X such that {W [A] : A ∈ R} is strongly locally finite (resp. strongly discrete).

Proof. As R is strongly locally finite (strongly dicrete), for each x ∈ X, there is some regular open set V_{x} containing x such that V_{x} ∩ R ≠ ∅ for at most finitely many (resp. one) members R of R. Then V = {V_{x} : x ∈ X} is a regular open cover of X and hence is regular even. Then by definition, there is some δ-nbd. U of Δ_{X} such that {U[x] : x ∈ X} < V. Again, U being a δ-nbd. of Δ_{X}, by Theorem 7, there is some symmetric δ-open nbd. W of Δ_{X} such that W ^{∘} W ⊆ U. Now, we observe that for any A ⊆ X,

(W ^{∘} W)[x] ∩ A = ∅ ⟹ W [x] ∩ W [A] = ∅.  (1)

In fact, y ∈ W [x] ∩ W [A] ⟹ (x, y) ∈ W and (z, y) ∈ W, for some z ∈ A ⟹ (x, y), (y, z) ∈ W (W being symmetric) ⟹ (x, z) ∈ W^{∘}W ⟹ z ∈ (W^{∘}W)[x]∩ A, establishing (1).

Now, for each x ∈ X, (W^{∘}W)[x] ⊆ U[x] ⊆ V, for some V ∈ V. By construction of V, V can meet at most finitely many members of R (resp. one member of R). Consequently, (W ^{∘} W)[x] can meet at most finitely many (resp. at most one) members A ∈ R. In view of (1), it follows that W [x] meets at most finitely many (resp. one) members like W [A], A ∈ R. Now W being a δ-open nbd. of Δ_{X}, W [x] is a δ-open nbd. of x, since W [x] = f_{x}^{−1}(W), where f_{x}: (X, τ_{s}) → (X × X, τ_{s} × τ_{s}), given by f_{x}(y) = (x, y) (for y ∈ X) is continuous. Thus for some regular open set T containing x, T ⊆ W [x] and clearly T meets at most finitely many (resp. one) members like W [A], A ∈ R. Hence, {W [A] : A ∈ R} is a strongly locally finite (resp. strongly discrete) collection.

Remark 9. If V is a regular open or a δ-open subset in X × X, then V [x] is δ-open in X, for each x ∈ X. So for any A ⊆ X, V [A] = ⋃_{x∈A} V [x] is also δ-open in X, and hence open in X. By the last theorem, each member of a strongly locally finite (or strongly discrete) family can be enlarged to an open set such that the latter collection is also strongly locally finite (resp. strongly discrete).

Corollary 10. Let X be a topological space such that each regular open cover of X is regular even. If each regular open cover U of X has a strongly locally finite (strongly discrete) refinement R, then each regular open cover U has a strongly locally finite (resp. strongly discrete) open refinement.

Proof. By Theorem 8, there is a δ-open nbd. V of the diagonal Δ_{X} such that {V [A] : A ∈ R} is strongly locally finite. This family may not be a refinement of U. For that, we choose U_{A} ∈ U, for each A ∈ R, such that A ⊆ U_{A}. Construct W_{A} = U_{A} ∩ V [A]. Then {W_{A} : A ∈ R} is the required family.

Theorem 11. If X is a topological space such that each regular open cover of X is regular even, then every regular open cover of X has an open σ-strongly discrete refinement (i.e. an open refinement which can be expressed as a countable union of strongly discrete families).

Proof. In view of Corollary 10, it is enough to find a σ-strongly discrete refinement of a given regular open cover U.

Since U is a regular even cover, there exists some δ-open nbd. V od Δ_{X} such that {V [x] : x ∈ X} < U. Define V_{0} = V. Then by induction, applying Theorem 7, we construct for each natural number n, a symmetric δ-open nbd. V_{n} of the diagonal Δ_{X} such that V_{n}^{∘}V_{n} ⊆ V_{n−1}. Let U_{1} = V_{1} and define inductively U_{n+1} = U_{n}^{∘}V_{n+1}, for n = 2, 3,…. It is easy to verify that U_{n} ⊆ V_{0} = V, for all natural numbers n. As {V_{0}[x] : x ∈ X} = {V [x] : x ∈ X} < U, it follows that {U_{n}[x] : x ∈ X} < U, for each natural number n. By Zermelo’s theorem choose some well-ordering < in X. For each natural number n and each x ∈ X, let U_{n}^{∗}(x) = U_{n}[x]\⋃{U_{n+1}[y] : y < x}. Now for each y ∈ X, V_{n+1}[U_{n}^{∗}(y)] is a δ-open nbd. of y. To verify that for each natural number n, the family U_{n} = {U_{n}^{∗}(x) : x ∈ X} is strongly discrete, it suffices to verify that U_{n}^{∗}(x)∩V_{n+1}[U_{n}^{∗}(y)] = ∅ whenever x ≠ y. In fact, suppose, if possible, p ∈ U_{n}^{∗}(x) ∩ V_{n+1}[U_{n}^{∗}(y)] (y ≠ x). Then p ∈ U_{n}^{∗}(x) = U_{n}[x] \ ⋃{U_{n+1}[y] : y < x} and p ∈ V_{n+1}[z], for some z ∈ U_{n}^{∗}(y). Thus

p ∈ U_{n}[x] and p ∉ U_{n+1}[y], for all y < x (y ∈ X),  (1)

and

(z, p) ∈ V_{n+1} for some z ∈ U_{n}[y] but z ∉ ⋃{U_{n+1}[q] : q < y}.  (2)

Then

(y, p) ∈ U_{n} ^{∘} V_{n+1} = U_{n+1}, i.e. p ∈ U_{n+1}[y].  (3)

From (1) and (3), we get y > x and consequently (by (2))

z ∉ U_{n+1}[x].  (4)

Again, (x, p) ∈ U_{n} and (p, z) ∈ V_{n+1} (since each V_{n} is symmetric) so that (x, z) ∈ U_{n} ^{∘} V_{n+1} = U_{n+1}, i.e. z ∈ U_{n+1}[x], a contradiction to (4). Thus our contention is established.

Now, {U_{n}[z] : z ∈ X} is a cover of X, for each n. We finally show that ⋃_{n=1}^{∞} U_{n} is a cover of X. Let x ∈ X. For each natural number n, choose the first point y(n) (say) of X such that x ∈ U_{n}[y(n)]. Let y(k) = min{y(n) : n = 1, 2,… }. Then x ∉ U_{n}[z], for z < y(k) and for each n. Put y = y(k). We see that x ∈ U_{k}[y] and x ∉ U_{k+1}[z] for all z < y. Thus x ∈ U_{k}^{∗}(y), proving that ⋃_{n=1}^{∞} U_{n} is a cover of X. This completes the proof.

Theorem 12. If each regular open cover of a space X has an open σ- (strongly) locally finite refinement, then each regular open cover of X has a (strongly) locally finite refinement.

Proof. Let U be a regular open cover of X. Let V = ⋃{V_{n} : n ∈ N} (where N stands for the set of all natural numbers) be an open σ- (strongly) locally finite refinement of U, where each V_{n} is (strongly) locally finite. For each n ∈ N, and each V ∈ V_{n}, let V_{n}^{∗} = V \ ⋃{U : U ∈ V_{k} for k < n} (resp. V_{n}^{∗} = int cl V \ ⋃{int cl U : U ∈ V_{k} for k < n}). Let W = {V_{n}^{∗} : n ∈ N, V ∈ V_{n}}. Let x ∈ X. If n is the first positive integer such that x ∈ W (resp. x ∈ int cl W) for some W ∈ V_{n}, then clearly x ∈ W_{n}^{∗}. Consequently, W is a cover of X. Obviously, W < U. Now to show that W is (strongly) locally finite, let x ∈ X, and n be the first positive integer such that V ∈ V_{n}, x ∈ V (resp. x ∈ int cl V). Then V (resp. int cl V) is an open (regular open) nbd. of x. We observe that for all k > n and for each P ∈ V_{k}, P_{k}^{∗} ∩ V = ∅ (resp. P_{k}^{∗} ∩ int cl V = ∅). So V can intersect at most the elements of V_{k}, for k ⩽ n. Now each V_{k} (k ⩽ n) is (strongly) locally finite. So there is a nbd. (regular open nbd.) U_{k} of x such that U_{k} can meet at mose finitely many members of V_{k} (k ⩽ n). Then S_{x} = V ∩U_{1} ∩U_{2} ∩⋯∩U_{n} (resp. S_{x} = int cl V ∩U_{1} ∩U_{2} ∩⋯∩U_{n}) is an open (regular open) nbd. of x such that S_{x} can meet at most finitely many members of ⋃_{n=1}^{∞} V_{n}, i.e., of V. Hence S_{x} can meet at most finitely many members of W, i.e., W is (strongly) locally finite.

Now, we are in a position to state the main theorem of this paper as follows.

Theorem 13. If X is an almost regular space, then the following are equivalent:

(i) X is nearly paracompact.

(ii) Each regular open cover of X has a strongly locally finite refinement. (iii) Each regular open cover of X has a locally finite refinement. (iv) Each regular open cover of X has a locally finite closed refinement.

(v) Each regular open cover of X has a locally finite regular closed refinement. (vi) Each regular open cover of X is regular even. (vii) Each regular open cover of X has an open σ-strongly discrete refinement. (viii) Each regular open cover of X has an open σ-strongly locally finite re-

finement.

(ix) Each regular open cover of X has an open σ-locally finite refinement.

Proof. (i) ⟹ (ii). Let U be any regular open cover of X. Then there is some open locally finite refinement V of U. For each x ∈ X, there exists some open nbd. U_{x} of x such that U_{x} ∩ V ≠ ∅, for at most finitely many V ∈ V, say V_{1}, V_{2}, …, V_{k}. Then cl U_{x} ∩V = ∅ for V ∈ V \{V_{1}, V_{2},…, V_{k}} and hence int cl U_{x} ∩V = ∅ for V ∈ V \ {V_{1}, V_{2},…, V_{k}}. Thus V is a strongly locally finite refinement of U.

The implications “(ii) ⟹ (iii)”, “(vii) ⟹ (viii)” and “(viii) ⟹ (ix)” are obvious, while the implications “(iii) ⟹ (iv)”, “(iv) ⟹ (v)”, “(v) ⟹ (vi)”, “(vi) ⟹ (vii)” and “(viii) ⟹ (ii)” follow from Theorems 5(i), 5(ii), 3, 11 and 12 respectively.

(ii) ⟹ (i). Let U be any regular open cover of X and R be its strongly locally finite refinement. As R < U, choose for each A ∈ R, some U_{A} ∈ U such that A ⊆ U_{A}. Now, X being almost regular, by Theorems 3 and 5, U is regular even. So, by Theorem 8, there is a δ-open nbd. V of the diagonal Δ_{X} in X × X such that {V [A] : A ∈ R} is strongly locally finite. Put W_{A} = V [A] ∩ U_{A}. Then {W_{A} : A ∈ R} is a locally finite open refinement of U.

Finally, we see that (ix) ⟹ (iii) is an immediate consequence of Theorem 12.

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(received 11.11.1997.)

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