����������� ������ 56 (2004), 63–76

UDK 515.122.2

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UNIFICATION OF SOME CONCEPTS SIMILAR TO THE LINDELÖF PROPERTY

T. Hatice Yalvaç

Abstract. In this paper the φ_{1,2}-Lindelöf property is defined and studied with the aim of unifying various concepts related to the Lindelöf property in General Topology.

1. Introduction

Unifications of various concepts in general topology have been studied in [1, 10–16,18, 21, 24–28]. Although the unifications discussed in [11] and [13] using operations were given for fuzzy topological spaces, it was pointed out that they apply equally well to topological spaces.

A principle aim of unification theory is to reduce the confusion caused by a plethora of very similar properties. For the most part we will not repeat here the definitions of the various properties used to illustrate the results obtained, although these will usually be clear from the particular choice of operations involved.

In a topological space (X, τ) we will use int, cl, scl, etc., to stand for the interior, closure and semi-closure operations, and so on. We will also use A^{o}, \overline{A} to stand for the interior and closure, respectively, of a subset A of X.

Definition 1.1. Let (X, τ) be a topological space. A mapping φ: P (X) → P (X) is called an operation on (X, τ) if A^{o} ⊂ φ(A) for all A ∈ P (X) and φ(∅) = ∅.

The class of all operations on a topological space (X, τ) will be denoted by O(X, τ).

A partial order “≤” on O(X, τ) is defined by φ_{1} ≤ φ_{2} ⇔ φ_{1}(A) ⊂ φ_{2}(A) for each A ∈ P (X). An operation φ ∈ O(X, τ) is called monotonous if φ(A) ⊂ φ(B) whenever A ⊂ B, (A, B ∈ P (X)).

Definition 1.2. Let φ ∈ O(X, τ) and A, B ⊂ X. Then A is called φ-open if A ⊂ φ(A). Likewise, B is called φ-closed if X \ B is φ-open. An operation φ̃ ∈ O(X, τ) is called the dual operation of φ if φ̃(A) = X \ φ(X \ A) for each A ∈ P (X).

AMS Subject Classification: 54 D 20, 54 A 20

Keywords and phrases: Lindelöf property, operation, unification, filter, convergence, supratopology.

If φ is monotonous, then the family of all φ-open sets is a supratopology (U ⊂ P (X)) is a supratopology on X means that ∅ ∈ U, X ∈ U and U is closed under arbitrary unions [2]).

Let (X, τ) be a topological space, φ ∈ O(X, τ), U ⊂ P (X) and x ∈ X. We will use the following notation.

U(x) = { U : x ∈ U ∈ U },
φO(X) = { U : U ⊂ X, U is φ-open },
φC(X) = { K : K ⊂ X, K is φ-closed },
φO(X, x) = { U : U ∈ φO(X), x ∈ U },
N (U, x) = { N : N ⊂ X and there exists a U ∈ U(x) such that U ⊂ N }.

Definition 1.3. Take φ ∈ O(X, τ) and U ⊂ P (X). Then φ is called regular with respect to (shortly w.r.t.) U if x ∈ X and given U, V ∈ U(x) there exists W ∈ U(x) such that φ(W) ⊂ φ(U) ∩ φ(V).

For any operation φ ∈ O(X, τ), τ ⊂ φO(X) and X, ∅ are both φ-open and φ-closed.

Definition 1.4. Let φ_{1}, φ_{2} ∈ O(X, τ), A ⊂ X. Then

a) x ∈ φ_{1,2} int A ⇔ there exists U ∈ φ_{1}O(X, x) such that φ_{2}(U) ⊂ A.

b) x ∈ φ_{1,2} cl A ⇔ for each U ∈ φ_{1}O(X, x), φ_{2}(U) ∩ A ≠ ∅.

c) A is φ_{1,2}-open ⇔ A ⊂ φ_{1,2} int A.

d) A is φ_{1,2}-closed ⇔ φ_{1,2} cl A ⊂ A. If A ⊂ B then φ_{1,2} int A ⊂ φ_{1,2} int B and φ_{1,2} cl A ⊂ φ_{1,2} cl B. Clearly, for

any set A we have X \ φ_{1,2} int A = φ_{1,2} cl(X \ A) and A is φ_{1,2}-open iff X \ A is φ_{1,2}-closed.

We will use φ_{1,2}O(X) (φ_{1,2}C(X)) to denote the family of all φ_{1,2}-open subsets (the family of all φ_{1,2}-closed subsets) of X.

Theorem 1.5. ([13]) Let φ_{1}, φ_{2} ∈ O(X, τ).

a) φ_{1,2}O(X) is a supratopology on X.

b) If φ_{2} is regular w.r.t. φ_{1}O(X) then φ_{1,2}O(X) is a topology on X and a subset K of X is closed w.r.t. this topology iff φ_{1,2} cl K ⊂ K.

c) If φ_{2} is regular w.r.t. φ_{1}O(X) and if φ_{2} ≥ ı or φ_{2} ≥ φ_{1} then φ_{1,2}O(X) is a topology on X and a set K is closed w.r.t. this topology iff φ_{1,2} cl K = K.

In [17], conditions were obtained under which the operator k: P (X) → P (X) defined by k(A) = φ_{1,2} cl A for each A ∈ P (X) is a Kuratowski closure operator.

Example 1.6. Let the following operations be defined on a topological space (X, τ): φ_{1} = int, φ_{2} = cl ∘ int, φ_{3} = cl, φ_{4} = scl, φ_{5} = ı (ı is the identity operation), φ_{6} = int ∘ cl. Then: φ_{1} ≤ φ_{2} ≤ φ_{3}, φ_{1} ≤ φ_{5} ≤ φ_{4} ≤ φ_{3} and φ_{1} ≤ φ_{6} ≤ φ_{4}. φ_{1}O(X) = τ, φ_{2}O(X) = SO(X) = the family of semi-open sets. φ_{3}O(X) = φ_{5}O(X) = φ_{4}O(X) = P (X) = the power set of X. φ_{6}O(X) = P O(X) = the family of pre-open sets. φ_{1,3}O(X) = τ_{θ} = the topology of all θ-open sets. φ_{2,4}O(X) = SθO(X) = the family of semi-θ-open sets. φ_{1,6}O(X) = τ_{s} = the semi regularization topology of X. This is the topology

with base RO(X) consisting of the regular open sets = the family of δ-open sets. φ_{2,3}O(X) = θSO(X) = the family of all θ-semi-open sets. φ_{1}, φ_{3} (φ_{2}, φ_{6}) are pairs of dual operations. Also, φ_{2}, φ_{3}, φ_{4}, φ_{5} and φ_{6} are

regular w.r.t. φ_{1}O(X).

We use SC(X) (P C(X), RC(X), SθC(X) and θSC(X), respectively) to denote the family of semi-closed (pre-closed, regular closed, semi-θ-closed, and θ-semi-closed) sets.

SR(X) = SO(X) ∩ SC(X) = the family of semi-regular sets. Clearly if φ_{1} is monotonous and φ_{2} = ı for φ_{1}, φ_{2} ∈ O(X, τ), then φ_{1,2}O(X) =

φ_{1}O(X) and φ_{1,2}C(X) = φ_{1}C(X).

2. Filters

Throughout this section the operations φ_{i}, i = 1, 2, … are defined on a topological space (X, τ).

Definition 2.1. ([26]) Let F be a filter (or filter-base) in (X, τ) and a ∈ X. Then F is said to:

a) φ_{1,2}-accumulate to a if a ∈ ⋂{φ_{1,2} cl F : F ∈ F};

b) φ_{1,2}-converge to a if for each U ∈ φ_{1}O(X, a), there exists an F ∈ F such that F ⊂ φ_{2}(U).

Theorem 2.2. ([26]) 1) A filter-base F_{b} φ_{1,2}-accumulates (φ_{1,2}-converges) to a iff filter generated by F_{b} φ_{1,2}-accumulates (φ_{1,2}-converges) to a.

2) If φ_{2} is monotonous, we can take the family N (φ_{1}O(X), a) instead of φ_{1}O(X, a) in the above definitions.

3) A filter F φ_{1,2}-converges to a iff {φ_{2}(U) : U ∈ φ_{1}O(X, a)} ⊂ F.

4) If F φ_{1,2}-converges to a then F φ_{1,2}-accumulates to a.

5) Let F ⊂ F′ for the filters F and F′

a) If F′ φ_{1,2}-accumulates to a, then F φ_{1,2}-accumulates to a.

b) If F φ_{1,2}-converges to a then F′ φ_{1,2}-converges to a.

6) If φ1^{′}O(X) ⊂ φ_{1}O(X) and φ2^{′} ≥ φ_{2} for the operations φ_{1}, φ_{2}, φ1^{′}, φ2^{′} ∈ O(X, τ), then a filter (or a filter-base) F φ1,2^{′}-accumulates (φ1,2^{′}-converges) to a whenever F φ_{1,2}-accumulates (φ_{1,2}-converges) to a.

A space (X, τ) is called φ_{1,2}-T_{2} if for each x, y ∈ X (x ≠ y) there are φ_{1}-open sets U_{x} and U_{y} such that x ∈ U_{x}, y ∈ U_{y} and φ_{2}(U_{x}) ∩ φ_{2}(U_{y}) = ∅ [13, 26].

Theorem 2.3. ([26]) Let (X, τ) be a φ_{1,2}-T_{2} space. If a filter F φ_{1,2}-converges to some point a ∈ X and φ_{1,2}-accumulates to some point b ∈ X then a = b.

Example 2.4. Let a ∈ X and F be a filter in (X, τ).

a) Let φ_{1} = cl ∘ int, φ_{2} = cl. F φ_{1,2}-converges (φ_{1,2}-accumulates) to a iff F rc-converges (rc-accumulates) to a since \{\overline{V} : V ∈ τ, x ∈ \overline{V} \} = \{\overline{U} : x ∈ U ∈ SO(X)} [9].

b) Let φ_{1} = int, φ_{2} = cl. F φ_{1,2}-converges to a iff F r-converges [8] (or equivalently almost converges [4]) to a. (X, τ) is φ_{1,2}-T_{2} iff (X, τ) is Urysohn.

c) Let φ_{1} = int, φ_{2} = ı. F φ_{1,2}-converges to a iff F converges to a in (X, τ). (X, τ) is φ_{1,2}-T_{2} iff (X, τ) is Hausdorff.

A filter F is called a σ-filter if F is closed under countable intersections and a filter-base F_{b} is called a σ-filter-base if F_{b} is a base of a σ-filter. Clearly in any topological space (X, τ), for any φ ∈ O(X, τ), F = {X} is a σ-filter and X ∈ φO(X).

Lemma 2.5. If φ_{2} ≥ φ_{1} or φ_{2} ≥ ı, then φ_{1}O(X) ⊂ φ_{2}O(X). The converse is false.

Example 2.6. Let φ_{1} = cl ∘ int, φ_{2} = semi- int be defined on ℝ with the usual topology. For A = (0, 1], φ_{1}(A) = [0, 1], φ_{2}(A) = (0, 1] and φ_{1}(A) ⊄ φ_{2}(A). i.e. φ_{1} ≰ φ_{2}.

For the set of rational numbers ℚ, ℚ ⊄ φ_{2}(ℚ) = ∅. i.e. φ_{2} ≱ ı. But φ_{1}O(ℝ) = SO(ℝ) = φ_{2}O(ℝ).

It can be easily seen that if φ_{2} is regular w.r.t. φ_{1}O(X) and φ_{1}O(X) ⊂ φ_{2}O(X) then φ_{1,2}O(X) is the same topology as that given in Theorem 1.5 (c).

Theorem 2.7. Let A ⊂ X. Then:

1) If there exists a σ-filter F containing A which φ_{1,2}-accumulates to a, then a ∈ φ_{1,2}clA.

2) If φ_{1}O(X) is closed under countable intersections and φ_{2} is monotonous, then,

a) a ∈ φ_{1,2} cl A iff there exists a σ-filter containing A which φ_{1,2}-converges to a.

b) A is φ_{1,2}-closed iff whenever there exists a σ-filter which contains A and φ_{1,2}-converges to a point a in X, then a ∈ A.

For the remainder of this section we will assume that φ_{1}O(X) is closed under countable intersections.

Theorem 2.8. If φ_{2} is monotonous then a σ-filter F φ_{1,2}-accumulates to a point a iff there exists a σ-filter F′ such that F ⊂ F′ and F′ φ_{1,2}-converges to a.

Theorem 2.9. a) If φ_{1}O(X) ⊂ φ_{2}O(X) then Φ_{a} = φ_{1}O(X, a) is σ-filterbase for each a ∈ X and Φ_{a} φ_{1,2}-converges to a.

b) If φ_{2} is monotonous and φ_{1}O(X) ⊂ φ_{2}O(X) then Φ_{a} = {φ_{2}(U) : U ∈ φ_{1}O(X, a)} is a σ-filter-base that φ_{1,2}-converges to a.

If φ_{2} is monotonous, then since φ_{1}O(X) is closed under countable intersections, clearly φ_{2} is regular w.r.t. φ_{1}O(X).

Theorem 2.10. Let φ_{2} be monotonous and φ_{1}O(X) closed under countable intersections. For any A ∈ P (X), let us define

cl^{∗} A = { x : there exists a σ-filter which contains A and φ_{1,2}-converges to x }.

Clearly cl^{∗} A = φ_{1,2} cl A.

a) The cl^{∗} operator defines the topology τ^{∗} given by

τ^{∗} = { U ⊂ X : (X \ U)^{∗} ⊂ X \ U }
= { U ⊂ X : φ_{1,2} cl(X \ U) ⊂ X \ U } = φ_{1,2}O(X).

b) If φ_{1}O(X) ⊂ φ_{2}O(X) then,

τ^{∗} = { U ⊂ X : (X \ U)^{∗} = X \ U }
= { U ⊂ X : φ_{1,2} cl A = A } = φ_{1,2}O(X).

c) If φ_{1}O(X) ⊂ φ_{2}O(X) and {φ_{2}(U) : U ∈ φ_{1}O(X)} ⊂ φ_{1,2}O(X) then cl^{∗} and φ_{1,2} cl are Kuratowski closure operators, since in this case φ_{1,2} cl(φ_{1,2} cl A) = φ_{1,2} cl A, for each A ∈ P (X).

Example 2.11. Let φ_{1} = int, φ_{2} = cl. Then:

φ_{1}O(X) = τ. φ_{2} is monotonous, regular w.r.t. φ_{1}O(X) and φ_{2} ≥ φ_{1}. Also, φ_{1,2} cl A = θ cl A for any A ∈ P (X) and cl^{∗} A = {x : there exists a σ-filter F such that A ∈ F and F φ_{1,2}-converges to x}.

If (X, τ) is a P space (i.e. each G_{δ} set is open), then a set A is θ-closed iff θ cl A = A iff whenever σ-filter contains A and φ_{1,2}-converges to a point a in X, then a ∈ A.

3. The Lindelöf property

Definition 3.1. Take φ_{1}, φ_{2} ∈ O(X, τ), X ∈ A ⊂ P (X) and A ⊂ X.

a) If A ⊂ ⋃ U for a subfamily U of A, then U is called an A-cover of A. If an A-cover U of A is countable (finite) then we call U a countable A-cover (finite A-cover) of A.

b) If each A-cover U of A has a countable subfamily U′ such that A ⊂ ⋃{φ_{2}(U) : U ∈ U′}, then we say that A is (A-φ_{2})-Lindelöf relative to X (shortly, a (A-φ_{2})-L. set)).

c) We call a (A-ı)-Lindelöf set relative to X a A-L. set for short.

d) If we take A = φ_{1}O(X) in (b), then we say A is φ_{1,2}-Lindelöf relative to X (shortly, a φ_{1,2}-L. set).

If we take A = φ_{1,2}O(X) in (c) we get the definition of a φ_{1,2}O(X)-L. set. If X is a φ_{1,2}-L. set relative to itself, then X will be called a φ_{1,2}-L. space. If X is a φ_{1,2}O(X)-L. set relative to itself, then X will be called a φ_{1,2}O(X)-L.

space.

In the sequel it will be assumed that all operations φ_{i}, i = 1, 2, … are defined on (X, τ) whenever they are used.

Theorem 3.2. If X ∈ A′ ⊂ A ⊂ P (X) and φ_{2} ≤ φ′_{2} then each (A-φ_{2})-L. set is an (A′-φ′_{2})-L. set. So if φ′_{1}O(X) ⊂ φ_{1}O(X) and φ_{2} ≤ φ′_{2} then a φ_{1,2}-L. set is a φ′_{1,2}-L. set.

Corollary 3.3. If φ′_{1} ≤ φ_{1} and φ_{2} ≤ φ′_{2}, then each φ_{1,2}-L. set is a φ′_{1,2}-L. set.

Remark 3.4. If φ_{2} is regular w.r.t. φ_{1}O(X), then a subset A of X is a φ_{1,2}O(X)-L. set iff A is Lindelöf relative to X in the topological space (X, φ_{1,2}O(X)).

Example 3.5. Let φ_{1} = cl ∘ int, φ_{2} = scl, φ′_{1} = int, φ′_{2} = int ∘ cl, φ′′_{1} = int and φ′′_{2} = cl. For a subset A, we have:

A is a φ_{1,2}-L. set iff A is a (SO(X)-scl)-L. set iff A is an SR-L. set (see Example 3.10 below).

An SR-L. space was called rs-Lindelöf in [7]. A is a φ_{1,2}O(X)-L. set iff A is a SθO(X)-L. set iff A is Lindelöf relative to X

in the supratopological space (X, SθO(X)).

Note that φ1,2^{′}-L. sets were defined for H(P) spaces under the name C_{1}-closed relative to X [23] and φ1,2^{′}-L. spaces were defined for H(P) spaces under the name strongly H(P)-closed spaces [23].

A is a φ1,2^{′}O(X)-L. set iff A is Lindelöf relative to X in (X, τ_{s}). The set A is a φ′′_{1,2}O(X)-L. set iff A is Lindelöf relative to X in (X, τ_{θ}). H(P)-closedness was defined for H(P) spaces. An H(P) space is called H(P)-

closed iff it is a φ′′_{1,2}-L. space [20].

Sets which are H(P)-closed relative to X were defined in H(P) spaces in [19]. So, in an H(P) space, a set A is H(P)-closed relative to X iff A is a φ′′_{1,2}-L. set.

A φ′′_{1,2}-L. space was called weakly Lindelöf in [5]. Since φ′′_{1} ≤ φ_{1} and φ_{2} ≤ φ′′_{2}, each φ_{1,2}-L. set is a φ′′_{1,2}-L. set. Hence each SR-L.

set is a (τ-cl)-L. set and each rs-Lindelöf space is a weakly Lindelöf space.

Theorem 3.6. If φ_{2} ≥ φ_{1} or φ_{2} ≥ ı, and if φ_{1} is monotonous, then each φ_{1}O(X)-L. set is a φ_{1,2}O(X)-L. set.

Proof. It can be easily seen that φ_{1,2}O(X) ⊂ φ_{1}O(X) under the given conditions [13], from which the proof is clear.

Theorem 3.7. Let B = {φ_{2}(U) : U ∈ φ_{1}O(X)}. Then:

a) If φ_{2}(U) ∈ φ_{1}O(X) and φ_{2}(φ_{2}(U)) ⊂ φ_{2}(U) for each U ∈ φ_{1}O(X) then B ⊂ φ_{1,2}O(X) ∩ φ_{1}O(X).

b) If φ_{1}O(X) ⊂ φ_{2}O(X) and B ⊂ φ_{1,2}O(X), then B is a base for the supratopology φ_{1,2}O(X).

c) If φ_{2} ≥ φ_{1} or φ_{2} ≥ ı, and if B ⊂ φ_{1,2}O(X), then B is a base for the supratopology φ_{1,2}O(X) [24].

d) If φ_{2} ≥ φ_{1} or φ_{2} ≥ ı, and if φ_{2}(U) ∈ φ_{1}O(X), φ_{2}(φ_{2}(U)) ⊂ φ_{2}(U) for each U ∈ φ_{1}O(X), then B is a base for the supratopology φ_{1,2}O(X) [24].

Proof. a) Let U ∈ φ_{1}O(X) and x ∈ φ_{2}(U). Then x ∈ φ_{2}(U) ∈ φ_{1}O(X) and φ_{2}(φ_{2}(U)) ⊂ φ_{2}(U), so x ∈ φ_{1,2} int φ_{2}(U). We have φ_{2}(U) ⊂ φ_{1,2} int φ_{2}(U). Hence φ_{2}(U) ∈ φ_{1,2}O(X) for each U ∈ φ_{1}O(X) and B ⊂ φ_{1,2}O(X) ∩ φ_{1}O(X).

b) Let A ∈ φ_{1,2}O(X) and x ∈ A. There exists a U ∈ φ_{1}O(X, x) such that φ_{2}(U) ⊂ A. Hence we have x ∈ U ⊂ φ_{2}(U) ⊂ A, φ_{2}(U) ∈ φ_{1,2}O(X) and φ_{2}(U) ∈ B.

The proofs of (c) and (d) are clear from Lemma 2.5 and (a), (b).

Theorem 3.8. The following results are valid.

1) If A is a φ_{1,2}-L. set, then it is a φ_{1,2}O(X)-L. set.

2) Let B ⊂ P (X). Then:

a) If X ∈ B ⊂ φ_{1,2}O(X), then each φ_{1,2}O(X)-L. set is a B-L. set.

b) If B is a base of φ_{1,2}O(X) then a set is a B-L. set iff it is a φ_{1,2}O(X)-L. set.

3) Let B = {φ_{2}(U) : U ∈ φ_{1}O(X)}. Then:

a) If φ_{1}O(X) ⊂ φ_{2}O(X) then each B-L. set is a φ_{1,2}-L. set.

b) If φ_{1}O(X) ⊂ φ_{2}O(X) and B ⊂ φ_{1,2}O(X) then a set is a φ_{1,2}-L. set iff it is a φ_{1,2}O(X)-L. set iff it is a B-L. set.

c) If φ_{2} ≥ φ_{1} or φ_{2} ≥ ı, and if B ⊂ φ_{1,2}O(X), then a set is a B-L. set iff it is a φ_{1,2}O(X)-L. set iff it is a φ_{1,2}-L. set.

d) If φ_{2} ≥ φ_{1} or φ_{2} ≥ ı, and if φ_{2}(U) ∈ φ_{1}O(X), φ_{2}(φ_{2}(U)) ⊂ φ_{2}(U) for each U ∈ φ_{1}O(X), then a set is a B-L. set iff it is a φ_{1,2}O(X)-L. set iff it is a φ_{1,2}-L. set.

Theorem 3.9. If φ_{2}(U) = φ_{3}(U) for each U ∈ φ_{1}O(X), then for a subset A, we have:

a) A is a φ_{1,2}-L. set iff it is a φ_{1,3}-L. set.

b) A is a φ_{1,2}O(X)-L. set iff it is a φ_{1,3}O(X)-L. set.

Example 3.10. a) Let φ_{1} = cl ∘ int, φ_{2} = scl. Then: φ_{1}O(X) = SO(X) and φ_{1,2}O(X) = SθO(X). For U ∈ φ_{1}O(X) = SO(X)

we have φ_{2}(U) = scl U ∈ SR(X) ⊂ SO(X). Also, φ_{2}(φ_{2}(U)) = scl(scl U) = scl U = φ_{2}(U) and φ_{2} ≥ ı. Thus, B = {scl U : U ∈ φ_{1}O(X)} = SR(X), whence B = SR(X) is a base for the supratopology φ_{1,2}O(X) = SθO(X).

So, a set is a SR-L. set iff it is a (SO(X)-scl)-L. set iff it is a SθO(X)-L. set.

b) Let φ_{1} = int, φ_{2} = int ∘ cl, φ_{3} = scl. Then: φ_{1}O(X) = τ, φ_{1,2}O(X) = φ_{1,3}O(X) = τ_{s}, φ_{2} ≥ φ_{1} and φ_{3} ≥ φ_{1}. For each

U ∈ φ_{1}O(X) = τ we have φ_{2}(U) = \overline{U}^{o} = U ∪ \overline{U}^{o} = scl U [3], and φ_{3}(U) = φ_{2}(U) ∈ φ_{1}O(X) = τ, φ_{2}(φ_{2}(U)) = \overline{(\overline{U}^{o})}^{o} = \overline{U}^{o} = φ_{2}(U). Then, B = \{φ_{2}(U) : U ∈ φ_{1}O(X)} = RO(X) and RO(X) is a base for τ_{s}.

So, a set is a RO(X)-L. set iff it is a (τ-int ∘ cl)-L. set iff it is a τ_{s}-L. set iff a (τ-scl)-L. set.

Remark 3.11. a) For φ_{2} = ı, the φ_{1,2}-L. property relative to X coincides with the φ_{1}O(X)-L. property relative to X.

b) If φ_{1} is monotonous and φ_{2} = ı, then for A ⊂ X we have: The set A is a φ_{1,2}-L. set iff A is a φ_{1}O(X)-L. set iff A is a φ_{1,2}O(X)-L. set. Proof. a) Clear.

b) If φ_{1} is monotonous then φ_{1}O(X) is a supratopology and for φ_{2} = ı we have φ_{1}O(X) = φ_{1,2}O(X), from which the result follows easily.

Theorem 3.12. Take X ∈ A ⊂ P (X), A ⊂ X and let K = {X \ U : U ∈ A}. Then the following are equivalent:

a) A is an A-L. set.

b) If W is any subfamily of K such that for each countable subfamily W′ of W we have A ∩ (⋂ W′) ≠ ∅, then A ∩ (⋂ W) ≠ ∅.

c) If W is any subfamily of K with A∩(⋂ W) = ∅ then there exists a countable subfamily W′ of W such that A ∩ (⋂ W′) = ∅.

Theorem 3.13. If ∅, X ∈ A ⊂ P (X), then X is an A-L. space iff for each U ∈ A, X \ U is a A-L. set.

If we take A = φ_{1,2}O(X) in Theorem 3.12 and Theorem 3.13, then K = φ_{1,2}C(X) and we obtain equivalent conditions for a set to be a φ_{1,2}O(X)-L. set, and equivalent conditions for the space X to be a φ_{1,2}O(X)-L. space.

Theorem 3.14. If φ̃_{2} is the dual of φ_{2} then the following are equivalent.

a) A is a φ_{1,2}-L. set.

b) For any family Φ of φ_{1}-closed sets with A ∩ (⋂ Φ) = ∅, there exists a countable subfamily Φ′ of Φ such that A ∩ (⋂{φ̃_{2}(F) : F ∈ Φ′}) = ∅.

c) If Φ is a family of φ_{1}-closed sets such that for each countable subfamily Φ′ of Φ we have A ∩ (⋂{φ̃_{2}(F) : F ∈ Φ′}) ≠ ∅, then A ∩ (∩Φ) ≠ ∅.

Theorem 3.15. Take A ⊂ X and B = {φ_{2}(U) : U ∈ φ_{1}O(X)}. If φ_{2} ≥ φ_{1} or φ_{2} ≥ ı, and if φ_{2}(U) ∈ φ_{1}O(X), φ_{2}(φ_{2}(U)) ⊂ φ_{2}(U) for each U ∈ φ_{1}O(X), then the following are equivalent: a) A is a φ_{1,2}-L. set.

b) A is a B-L. set.

c) A is a φ_{1,2}O(X)-L. set.

d) If W is any subfamily of {X \ φ_{2}(U) : U ∈ φ_{1}O(X)} such that for each countable subfamily W′ of W we have A ∩ (⋂ W′) ≠ ∅ then A ∩ (⋂ W) ≠ ∅.

e) If W is any subfamily of {X \ φ_{2}(U) : U ∈ φ_{1}O(X)} with A ∩ (⋂ W) = ∅ then there exists a countable subfamily W′ of W such that A ∩ (⋂ W′) = ∅.

f) If W is any subfamily of {X\U : U ∈ φ_{1,2}O(X)} such that for each countable subfamily W′ of W we have A ∩ (⋂ W′) ≠ ∅, then A ∩ (∩W) ≠ ∅.

g) If W is any subfamily of {X \ U : U ∈ φ_{1,2}O(X)} such that A ∩ (⋂ W) = ∅, then there exists a countable subfamily W′ of W such that A ∩ (⋂ W′) = ∅.

If φ̃_{2} is the dual of φ_{2} then the following statements (h) and (i) are equivalent to each one of the above statements.

h) For any family of Φ of φ_{1}-closed sets with A ∩ (⋂ Φ) = ∅, there exists a countable subfamily Φ′ of Φ such that A ∩ (⋂{φ̃_{2}(F) : F ∈ Φ′}) = ∅.

i) If Φ is any family of φ_{1}-closed sets such that for each countable subfamily Φ′ of Φ we have A ∩ (⋂{φ̃_{2}(F) : F ∈ Φ′}) ≠ ∅, then A ∩ (⋂ Φ) ≠ ∅.

Now, using the equality φ_{1,2}C(X) = {X \ U : U ∈ φ_{1,2}O(X)} and the Theorems 3.13 and 3.15 we obtain the following theorem.

Theorem 3.16. Under the hypotheses of Theorem 3.15 the following are equivalent.

a) X is a φ_{1,2}-L. space.

b) X is a B-L. space.

c) X is a φ_{1,2}O(X)-L. space.

d) For each U ∈ φ_{1}O(X), X \ φ_{2}(U) is a φ_{1,2}-L. set.

e) For each U ∈ φ_{1}O(X), X \ φ_{2}(U) is a φ_{1,2}O(X)-L. set.

f) For each U ∈ φ_{1}O(X), X \ φ_{2}(U) is a B-L. set.

g) Each φ_{1,2}-closed set is a φ_{1,2}-L. set.

h) Each φ_{1,2}-closed set is a φ_{1,2}O(X)-L. set.

i) Each φ_{1,2}-closed set is a B-L. set. Other equivalent expressions may be obtained by taking X in place of A in

(d–i) of Theorem 3.15.

Example 3.17. Let φ_{1} = semi- int, φ_{2} = scl. Then: The conditions of Theorem 3.15 are satisfied and φ_{2} is the dual of φ_{1}. φ_{1}O(X) = SO(X), φ_{1}C(X) = SC(X), B = {φ_{2}(U) : U ∈ φ_{1}O(X)} = {scl U :

U ∈ SO(X)} = SR(X).

{X \ φ_{2}(U) : U ∈ φ_{1}O(X)} = SR(X). φ_{1,2}O(X) = SθO(X) and φ_{1,2}C(X) = SθC(X).

Now we obtain equivalent conditions for a set to be a (SO(X)-scl)-L. set by using Theorem 3.15, and equivalent conditions for the space (X, τ) to be a (SO(X)- scl)-L. space (equivalently, SR-Lindelöf) by using Theorem 3.16.

Theorem 3.18. Let (X, τ) be a topological space and A ⊂ X. Then the following are equivalent.

a) For each SO(X)-cover U of A, there exists a countable subfamily U′ of U such that A ⊂ ∪{scl U : U ∈ U′}.

b) For each SR(X)-cover U of A there exists a countable subfamily U′ of U such that A ⊂ ⋃ U′.

c) For each SθO(X)-cover U of A, there exists a countable subfamily U′ of A such that A ⊂ ⋃ U′.

d) If W is a subfamily of SR(X) such that for each countable subfamily W′ of W, we have A ∩ (⋂ W′) ≠ ∅, then A ∩ (⋂ W) ≠ ∅.

e) If W is any subfamily of SR(X) such that A ∩ (⋂ W) = ∅, then there exists a countable subfamily W′ of W such that A ∩ (⋂ W′) = ∅.

f) If W is a subfamily of SθC(X) such that for each countable subfamily W′ of W we have A ∩ (⋂ W′) ≠ ∅, then A ∩ (⋂ W) ≠ ∅.

g) If W is any subfamily of SθC(X) with A ∩ (⋂ W) = ∅ then there exists a countable subfamily W′ of W such that A ∩ (⋂ W′) = ∅.

h) For any subfamily Φ of semi-closed sets with A ∩ (⋂ Φ) = ∅ there exists a countable subfamily Φ′ of Φ such that A ∩ (⋂{semi- int(F) : F ∈ Φ′}) = ∅.

i) If Φ is any family of semi-closed sets such that for each countable subfamily Φ′ of Φ, we have A ∩ (⋂{semi- int(F) : F ∈ Φ′}) ≠ ∅, then A ∩ (⋂ Φ) ≠ ∅.

For a topological space (X, τ) to be rs-Lindelöf [7] (equivalently, a (SO(X)- scl)-L. space), we can add the following equivalent statements to those given in Theorem 3.18, by taking X instead of A there.

j) Each semi-regular set is a SR(X)-L. set.

k) Each semi-regular set is a SθO(X)-L. set.

l) Each semi-regular set is a (SO(X)-scl)-L. set.

m) Each φ_{1,2}-closed set is a SR(X)-L. set.

n) Each φ_{1,2}-closed set is a SθO(X)-L. set.

o) Each φ_{1,2}-closed set is a (SO(X)-scl)-L. set.

Example 3.19. a) Let φ_{1} = cl ∘ int, φ_{2} = cl. Then: The conditions of Theorem 3.15 are satisfied. φ̃_{2} = int is the dual of φ_{2}.

φ_{1}O(X) = SO(X) and φ_{1}C(X) = SC(X).

B = {φ_{2}(U) : U ∈ φ_{1}O(X)} = {cl U : U ∈ SO(X)} = RC(X). φ_{1,2}O(X) = θSO(X) and φ_{1,2}C(X) = θSC(X) = {X \ U : U ∈ φ_{1,2}O(X)}.

{X \ B : B ∈ B} = RO(X).

A set A is a φ_{1,2}-L. set iff it is a (SO(X)-cl)-L. set iff it is a θSO(X)-L. set iff it is RC(X)-L. set.

Hence we obtain, (X, τ) is a φ_{1,2}-L. space iff X is rc-Lindelöf (rc-Lindelöf spaces were defined by D. Janković and Ch. Konstadilaki, as mentioned in [5]).

Again, we can get equivalent conditions for a set to be a (SO(X)-cl)-L. set in a topological space (X, τ), and equivalent conditions for a space to be rc-Lindelöf by using Theorems 3.15 and 3.16.

b) Let φ_{1} = int, φ_{2} = int ∘ cl on (X, τ). Then: The conditions of Theorem 3.15 are satisfied and φ̃_{2} = cl ∘ int. φ_{1}O(X) = τ, φ_{1}C(X) = family of closed sets. φ_{1,2}O(X) = τ_{s} = family of

δ-open sets, {X \ U : U ∈ φ_{1,2}O(X)} = φ_{1,2}C(X) = family of τ_{s}-closed sets = family of δ-closed sets.

B = \{φ_{2}(U) : U ∈ φ_{1}O(X)\} = \{\overline{U}^{o} : U ∈ τ\} = RO(X). \{X \ B : B ∈ B\} = {X \ φ_{2}(U) : U ∈ φ_{1}O(X)} = RC(X).

Now, using Theorems 3.15 and 3.16 we can obtain equivalent conditions for a set to be a (τ-int ∘ cl)-L. set, and equivalent conditions for a space to be a (τint ∘ cl)-L. space. Also, in H(P) spaces, equivalent conditions for a set to be C_{1}- closed relative to the space and equivalent conditions for a space to be strongly H(P)-closed.

Some equalities related to different types of closure can be obtained using operations. Some of them were given in [24] and some of them are given below. In a topological space (X, τ) we have:

1) \overline{T} = θ-clT = τ_{s}-cl T, for each T ∈ τ.

2) θ-semi-cl T = scl T = semi-θ-cl T, for each T ∈ τ.

3) \overline{T} = τ_{s}-cl T = τ^{α}-cl T, for each T ∈ τ^{α}, (τ^{α} = \{U : U ⊂ X, U ⊂ \overline{U^{o}}^{o}\}).

4) semi-θ-cl T = scl T, for each T ∈ τ^{α}. Example 3.20. Let φ_{1} = int, φ_{2} = τ-cl, φ_{3} = θ-cl, φ_{4} = τ_{s}-cl. Then: φ_{1}O(X) = τ. For each U ∈ τ we have φ_{2}(U) = φ_{3}(U) = φ_{4}(U). φ̃_{2} = τ-int, for each φ_{1}-closed set K, int K = θ-int K = τ_{s}-int K. φ_{1,2}O(X) = φ_{1,3}O(X) = φ_{1,4}O(X) = τ_{θ}. {X \ U : U ∈ φ_{1,2}O(X)} = φ_{1,2}C(X) = φ_{1,3}C(X) = φ_{1,4}C(X) = family of

θ-closed sets.

Now, by using Theorems 3.9, 3.12, 3.13, 3.14 and 3.16 we can add some more equivalent conditions for a set to be a φ_{1,2}-L. set (φ_{1,2}O(X)-L. set), and for a space (X, τ) to be a φ_{1,2}-L. space (φ_{1,2}O(X)-L. space) for some special choices of the operations φ_{1} and φ_{2}.

Lemma 3.21. ([25]) For each U ∈ φ_{1}O(X) we have U ⊂ φ_{1,2} int φ_{2}(U).

Theorem 3.22. The following are equivalent for any subset A of X.

a) A is a φ_{1,2}-L. set. b) Each σ-filter-base in A φ_{1,2}-accumulates in X to some point in A.

c) Every σ-filter-base in X which meets A φ_{1,2}-accumulates in X to some point of A.

d) For any family of non-empty sets W with A ∩ (⋂{φ_{1,2} cl F : F ∈ W}) = ∅, there exists a countable subfamily W′ of W such that A ∩ (⋂{F : F ∈ W′}) = ∅.

e) For any family W of non-empty sets such that for each countable subfamily W′ of W we have A ∩ (⋂{F : F ∈ W′}) ≠ ∅, it holds that A ∩ (⋂{φ_{1,2} cl F : F ∈ W}) ≠ ∅.

f) If F is a σ-filter-base such that A ∩ (⋂{φ_{1,2} cl F : F ∈ F}) = ∅, then there exists an F ∈ F such that F ∩ A = ∅.

Proof. (a ⇒ b). Let A be a φ_{1,2}-L. set. Let us accept that some σ-filter-base F_{b} in A does not accumulate to any point in A.

Then, for each a ∈ A there exists a U_{a} ∈ φ_{1}O(X, a) and an F_{a} ∈ F_{b} such that F_{a} ∩ φ_{2}(U_{a}) = ∅. Since A is a φ_{1,2}-L. set there is a countable subset C of A such that A ⊂ ⋃_{a∈C} φ_{2}(U_{a}). Hence (⋂_{a∈C} F_{a}) ∩ (⋃_{a∈C} φ_{2}(U_{a})) = ∅. Hence there exists an F ∈ F_{b} such that F ⊂ ⋂_{a∈C} F_{a}. Thus F ∩ A = ∅. This contradiction completes the proof.

(b ⇒ c). If F_{b} is a σ-filter-base in X which meets A, then F′_{b} = {A∩F : F ∈ F_{b}} is a σ-filter-base in A and hence in X.

The σ-filter F′ generated by F′_{b} contains A and F_{b} ⊂ F′. From (a), F′_{b} φ_{1,2}- accumulates to some point a in A. So F_{b} φ_{1,2}-accumulates to a.

(c ⇒ d). Let {F_{i} : i ∈ I} be a family of non-empty sets with (∩{φ_{1,2} cl F_{i} : i ∈ I}) ∩ A = ∅. Let us accept that for each countable subset J of I we have (⋂_{j∈J} F_{j}) ∩ A ≠ ∅. Then F_{b} = {⋂_{j∈J} F_{j} : J ⊂ I, J countable } is a σ-filter-base in X which meets A. Hence there exists a point a in A such that F_{b} φ_{1,2}-accumulates to a.

For each U ∈ φ_{1}O(X, a) and for each F ∈ F_{b}, F ∩ φ_{2}(U) ≠ ∅. So for each U ∈ φ_{1}O(X, a) and for each i ∈ I, F_{i} ∩φ_{2}(U) ≠ ∅. This gives a ∈ (⋂_{i∈I} φ_{1,2} cl F_{i})∩ A ≠ ∅. This contradiction completes the proof.

(d ⇒ a). Let A ⊂ ∪U where U ⊂ φ_{1}O(X). Then (⋂{X \ U : U ∈ U}) ∩ A = ∅. Let K = {X \φ_{2}(U) : U ∈ U}. If there exists a U ∈ U such that X \φ_{2}(U) = ∅,

then A ⊂ φ_{2}(U) = X. On the other hand, if for each U ∈ U, X \ φ_{2}(U) ≠ ∅ let us show that (⋂{φ_{1,2} cl(X \ φ_{2}(U)) : U ∈ U}) ∩ A = ∅.

Firstly, let us see that for each U ∈ U we have φ_{1,2} cl(X \ φ_{2}(U)) ⊂ X \ U. Let U ∈ U and x ∈ φ_{1,2} cl(X \ φ_{2}(U). From Lemma 3.20, U ⊂ φ_{1,2} int φ_{2}(U). Now x ∈ φ_{1,2} cl(X \ φ_{2}(U)) = (X \ φ_{1,2} int φ_{2}(U)) ⊂ X \ U. Hence

(⋂{φ_{1,2} cl(X \ φ_{2}(U) : U ∈ U}) ∩ A ⊂ (⋂{X \ U : U ∈ U}) ∩ A = ∅.

From (d) there exists a countable subfamily U′ of U such that (⋂{X \ φ_{2}(U) : U ∈ U′}) ∩ A = ∅. Hence A ⊂ ∪{φ_{2}(U) : U ∈ U′}, so A is a φ_{1,2}-L. set.

d ⇔ e, d ⇒ f and f ⇒ c are clear.

Theorem 3.23. If (X, τ) is φ_{1,2}-T_{2}, φ_{1}O(X) is closed under countable intersections and φ_{2} is monotonous then each φ_{1,2}-L. set is φ_{1,2}-closed.

Proof. Let x ∈ φ_{1,2} cl A. Then there exists a σ-filter F such that A ∈ F and F φ_{1,2}-converges to x. Since F meets A and A is a φ_{1,2}-L. set there exists a point a in A such that F φ_{1,2}-accumulates to a.

From Theorem 2.3 we must have a = x, so x ∈ A since we have φ_{1,2} cl A ⊂ A, and A is a φ_{1,2}-closed set.

Example 3.24. Let φ_{1} = int and φ_{2} ∈ O(X, τ) be a monotonous operation. Then:

φ_{1}O(X) = τ. φ_{1}O(X) is closed under countable intersections iff (X, τ) is a P space.

So, using σ-filters in P spaces we can obtain equivalent characterizations for a φ_{1,2}-L. set.

If (X, τ) is an H(P)-space, then by choosing φ_{1} = int and φ_{2} = cl, we get equivalent conditions for a space to be H(P)-closed.

Proceeding in this way we can obtain many known results, some of which occur in [4,7,20,22], and also many new results.

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(received 25.01.2001, in revised form 10.09.2004)

Hacettepe University, Mathematics Department, 06532 Beytepe Ankara, Turkey

 E-mail: hayal@hacettepe.edu.tr