����������� ������ 56 (2004), 77–80

UDK 515.122.2

���������� ������ ��� research paper

SEMIREGULAR PROPERTIES AND GENERALIZED LINDELÖF SPACES

A. J. Fawakhreh and A. Kılıçman

Abstract. Let (X, τ) be a topological space and (X, τ^{∗}) its semiregularization. Then a topological property P is semiregular provided that τ has property P if and only if τ^{∗} has the same property. In this work we study semiregular property of almost Lindelöf, weakly Lindelöf, nearly regular-Lindelöf, almost regular-Lindelöf and weakly regular-Lindelöf spaces. We prove that all these topological properties, on the contrary of Lindelöf property, are semiregular properties.

1. Introduction

Among various covering properties of topological spaces a lot of attention has been made to those covers which involve regularly open sets and regularly closed sets. In 1959 Frolik [5] introduced the notion of weakly Lindelöf space that was afterwards studied by several authors. In 1982 Balasubramanian [1] introduced and studied the notion of nearly Lindelöf spaces as a generalization of the nearly compact spaces defined by Singal and Mathur [8]. Then in 1986 Mršević et al.

[7] gave some characterizations of the classes of nearly Lindelöf spaces. In 1984 Willard and Dissanayake [9] gave the notion of almost Lindelöf spaces and in 1996 Cammaroto and Santoro [4] introduced the notion of nearly regular-Lindelöf, almost regular-Lindelöf and weakly regular-Lindelöf spaces by using the regular covers which were introduced by Cammaroto and Lo Faro [3] in 1981. The purpose of this paper is to prove that all these generalizations of Lindelöf spaces, which are topological properties, are semiregular properties.

Throughout this paper, a space X means a topological space (X, τ) on which no separation axioms are assumed unless explicitly stated otherwise. The interior and the closure of any subset A of (X, τ) will be denoted by Int(A) or Int_{τ} (A) and Cl(A) or Cl_{τ} (A) respectively. Recall that a subset A ⊆ X is called regularly open (regularly closed) if A = Int(Cl(A)) (A = Cl(Int(A)). The class of all regular open sets in a space (X, τ) is a base for a coarser topology than τ on X, called the semiregularization topology of X which is denoted by (X, τ^{∗}). Semiregularization topologies were studied by Mršević et al. [6] in 1985. If τ = τ^{∗} then X is said to be semiregular. The following results, to be used in the sequel, are known and easily can be proved.

AMS Subject Classification: 54 A 05, 54 A 20, 54 F 45

Keywords and phrases: Semiregular property, semiregularization topology, nearly Lindelöf, almost Lindelöf, weakly Lindelöf, nearly regular-Lindelöf, almost regular-Lindelöf and weakly regular-Lindelöf spaces.

Lemma 1.1. Let (X, τ) be a topological space and (X, τ^{∗}) its semiregularization. Then

(a) regularly open sets of (X, τ) are the same as regularly open sets of (X, τ^{∗}).

(b) regularly closed sets of (X, τ) are the same as regularly closed sets of (X, τ^{∗}).

(c) Int_{τ} (C) = Int_{τ^{∗}}(C) for any regularly closed set C.

(d) Cl_{τ} (A) = Cl_{τ^{∗}}(A) for every A ∈ τ.

Definition 1.1. (see [2]) Let (X, τ) be a topological space and (X, τ^{∗}) its semiregularization. A topological property P is called semiregular provided that τ has the property P if and only if τ^{∗} has the property P.

2. Semiregularization of Lindelöf spaces

Definition 2.1. (see [1], [9] and [5]). A topological space X is said to be nearly Lindelöf, almost Lindelöf and weakly Lindelöf if, for every open cover {U_{α} : α ∈ Δ} of X, there exists a countable subset {α_{n} : n ∈ N} ⊆ Δ such that

X = ⋃_{n∈N} Int(Cl(U_{α_{n}})), X = ⋃_{n∈N} Cl(U_{α_{n}}) and X = Cl( ⋃_{n∈N} U_{α_{n}}),

respectively.

Note first that a space (X, τ) is nearly Lindelöf if and only if (X, τ^{∗}) is Lindelöf (see [7], Theorem 1). Moreover, the Lindelöf property is not a semiregular property (see [7], Example 1). Note also that, in distinction to the Lindelöf property, the nearly Lindelöf property is a semiregular property (see [7], Corollary 4). Next we prove that almost Lindelöf and weakly Lindelöf properties are semiregular properties as we conclude from the following theorems.

Theorem 2.1. Let (X, τ) be a topological space. Then (X, τ) is almost Lindelöf if and only if (X, τ^{∗}) is almost Lindelöf.

Proof. Suppose that (X, τ) is almost Lindelöf and let {U_{α} : α ∈ Δ} be an open cover of (X, τ^{∗}), thus an open cover of the almost Lindelöf space (X, τ) since τ^{∗} ⊆ τ. So there exists a countable subset {α_{n} : n ∈ N} ⊆ Δ such that X = ⋃_{n∈N} Cl_{τ} (U_{α_{n}}). By Lemma 1.1(d), X = ⋃_{n∈N} Cl_{τ^{∗}}(U_{α_{n}}). This implies that (X, τ^{∗}) is almost Lindelöf.

Conversely, suppose that (X, τ^{∗}) is almost Lindelöf and let {V_{α} : α ∈ Δ} be an open cover of (X, τ). Then {Int_{τ} (Cl_{τ} (V_{α})) : α ∈ Δ} is an open cover of the almost Lindelöf space (X, τ^{∗}). Thus it has a countable subfamily {Int_{τ} (Cl_{τ} (V_{α_{n}})) : n ∈ N} such that

X = ⋃_{n∈N} Cl_{τ^{∗}}(Int_{τ} (Cl_{τ} (V_{α_{n}}))) = ⋃_{n∈N} Cl_{τ} (Int_{τ} (Cl_{τ} (V_{α_{n}}))) = ⋃_{n∈N} Cl_{τ} (V_{α_{n}}).

This shows that (X, τ) is almost Lindelöf and completes the proof.

Corollary 2.1. Almost Lindelöf property is a semiregular property.

Theorem 2.2. Let (X, τ) be a topological space. Then (X, τ) is weakly Lindelöf if and only if (X, τ^{∗}) is weakly Lindelöf.

Proof. The proof is quite similar to the proof of Theorem 2.1 above on using the fact that

Cl_{τ^{∗}}( ⋃_{n∈N} Int_{τ} (Cl_{τ} (V_{α_{n}}))) ⊆ Cl_{τ^{∗}}( ⋃_{n∈N} V_{α_{n}}) = Cl_{τ} ( ⋃_{n∈N} V_{α_{n}}).

Thus we choose to omit the details.

Corollary 2.2. Weakly Lindelöf property is a semiregular property.

3. Semiregularization of regular-Lindelöf Spaces

Recall that an open cover {U_{α} : α ∈ Δ} of a topological space X is called regular [3] if, for every α ∈ Δ, there exists a nonempty regularly closed subset C_{α} of X such that C_{α} ⊆ U_{α} and X = ⋃_{α∈Δ} Int(C_{α}). The following definition was given in [4] and is using regular covers to give some generalizations of Lindelöf spaces.

Definition 3.1. A topological space X is called weakly regular-Lindelöf (resp. almost regular-Lindelöf, nearly regular-Lindelöf) if every regular cover {U_{α} : α ∈ Δ} of X admits a countable subfamily {U_{α_{n}} : n ∈ N} such that

X = Cl( ⋃_{n∈N} U_{α_{n}}) ( resp. X = ⋃_{n∈N} Cl(U_{α_{n}}), X = ⋃_{n∈N} Int(Cl(U_{α_{n}}))).

The following implications are obvious from the definitions.

Lindelöf ⇒ nearly Lindelöf ⇒ almost Lindelöf ⇒ weakly Lindelöf
⇓  ⇓  ⇓
nearly  almost  weakly
regular-Lindelöf ⇒ regular-Lindelöf ⇒ regular-Lindelöf

Now we prove that the nearly regular-Lindelöf property is a semiregular property depending on Lemma 1.1 above.

Theorem 3.1. Let (X, τ) be a topological space. Then (X, τ) is nearly regular-Lindelöf if and only if (X, τ^{∗}) is nearly regular-Lindelöf.

Proof. Suppose that (X, τ) is nearly regular-Lindelöf and let {U_{α} : α ∈ Δ} be a regular cover of (X, τ^{∗}). Since τ^{∗} ⊆ τ and by Lemma 1.1 (a, b and d) we have, {U_{α} : α ∈ Δ} is a regular cover of the nearly regular-Lindelöf space (X, τ). So it has a countable subset {U_{α_{n}} : n ∈ N} such that X = ⋃_{n∈N} Int_{τ} (Cl_{τ} (U_{α_{n}})) = ⋃_{n∈N} Int_{τ^{∗}}(Cl_{τ^{∗}}(U_{α_{n}})) (by Lemma 1.1(c,d)). This implies that (X, τ^{∗}) is nearly regular-Lindelöf.

Conversely, suppose that (X, τ^{∗}) is nearly regular-Lindelöf and let {U_{α} : α ∈ Δ} be a regular cover of (X, τ). Then since U_{α} ⊆ Int_{τ} (Cl_{τ} (U_{α})), Lemma 1.1 above implies that {Int_{τ} (Cl_{τ} (U_{α})) : α ∈ Δ} is a regular cover of the nearly regular-Lindelöf space (X, τ^{∗}). Thus there exists a countable subfamily {α_{n} : n ∈ N}, such that X = ⋃_{n∈N} Int_{τ^{∗}}(Cl_{τ^{∗}}(Int_{τ} (Cl_{τ} (U_{α_{n}})))) = ⋃_{n∈N} Int_{τ^{∗}}(Cl_{τ^{∗}}(U_{α_{n}})) = ⋃_{n∈N} Int_{τ} (Cl_{τ} (U_{α_{n}})) (by Lemma 1.1(a,c,d)). This implies that (X, τ) is a nearly regular-Lindelöf space and completes the proof.

Corollary 3.1. Nearly regular-Lindelöf property is a semiregular property. Next we prove that the almost regular-Lindelöf and weakly regular-Lindelöf

properties are semiregular properties.

Theorem 3.2. Let (X, τ) be a topological space. Then (X, τ) is almost regular-Lindelöf if and only if (X, τ^{∗}) is almost regular-Lindelöf.

Proof. The proof is similar to the proof of Theorem 3.1 above, thus we choose to omit the details.

Corollary 3.2. Almost regular-Lindelöf property is a semiregular property. Theorem 3.3. Let (X, τ) be a topological space. Then (X, τ) is weakly regular-

Lindelöf if and only if (X, τ^{∗}) is weakly regular-Lindelöf.

Proof. The proof is similar to the proof of Theorem 3.1 above on using the fact that Cl_{τ} (⋃_{n∈N}(Int_{τ} (Cl_{τ} (U_{α_{n}}))) ⊆ Cl_{τ} (⋃_{n∈N} U_{α_{n}}) = Cl_{τ^{∗}}(⋃_{n∈N} U_{α_{n}}). Thus the details are omitted.

Corollary 3.3. Weakly regular-Lindelöf property is a semiregular property.

REFERENCES

[1] Balasubramanian, G., On some generalizations of compact spaces, Glasnik Math., 17(37) (1982), 367–380.

[2] Cameron, D. E., Properties of S-closed spaces, Proc. Amer. Math. Soc., 72 (1978), 581–586.

[3] Cammaroto, F. and Lo Faro, G., Spazi weakly compact, Riv. Mat. University Parma, 14(7) (1981), 383–395.

[4] Cammaroto, F. and Santoro, G., Some counterexamples and properties on generalizations of Lindelöf spaces, Internat. J. Math. and Math. Sci., 19(4) (1996), 737–746.

[5] Frolik, Z., Generlization of compact and Lindelöf spaces, Czechoslovak Math. J., 9(84) (1959), 172–217.

[6] Mršević, M., Reilly, I. L. and Vamanamurthy, M. K., On semi-regularization topologies, J. Australian Math. Soc. (series A), 38 (1985), 40–54.

[7] Mršević, M., Reilly, I. L. and Vamanamurthy, M. K., On nearly Lindelöf spaces, Glasnik Mat., 21(41) (1986), 407–414.

[8] Singal, M. K. and Mathur, A., On nearly compact spaces, Boll. Un. Mat. Ital., 4(2) (1969), 702–710.

[9] Willard, S and Dissanayake, U.N.B., The almost Lindelöf degree, Canad. Math. Bull., 27(4) (1984), 452–455.

(received 01.08.2001, in revised form 15.10.2004)

A. J. Fawakhreh, Philadelphia University, Department of Basic Sciences, 19392, Jordan, E-mail:

 abuanas7@hotmail.com

A. Kılıçman, University Putra Malaysia, Department of Mathematics, 43400 UPM, Serdang, Selangor, Malaysia, E-mail: akilic@fsas.upm.edu.my