����������� ������ 56 (2004), 1–12

UDK 515.122.2

���������� ������ ��� research paper

A UNIFIED THEORY OF CONTINUITY

T. Hatice Yalvaç

Abstract. This paper further develops the theory of operations on a topological space with the aim of producing a uniform framework for the study of generalized forms of continuity. To illustrate the utility of this approach the results obtained are used to obtain many new and known characterizations of strong Θ-semi-continuity, weak continuity and almost continuity.

1. Introduction

Various unification theories in topological spaces have been studied by several authors [1, 5–11, 14, 15]. This paper develops the theory of operations on a topological space with the aim of producing a uniform framework for the study of generalized forms of continuity.

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

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).

The operations φ, φ̃ are said to be dual if φ(A) = X \ (φ̃(X \ A)) (equivalently φ̃(A) = X \ (φ(X \ A))) for each A ∈ P (X).

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), U ⊂ P (X) and U(x)= { U : U ∈ U, x ∈ U } for x ∈ X. Then φ is called: a) regular with respect to (shortly, w.r.t.) U if for each x ∈ X and U, V ∈ U(x),

AMS Subject Classification: 54 C 08

Keywords and phrases: Continuity, operation, unification, supratopology

there exists a W ∈ U(x) such that φ(W) ⊂ φ(U) ∩ φ(V).

b) weakly finite intersection preserving (shortly, W.F.I.P.) w.r.t U if whenever U ∈ U, A ∈ P (X), we have U ∩ φ(A) ⊂ φ(U ∩ A).

Remark 1.3. Monotonicity of φ ∈ O(X, τ) is not sufficient for regularity of φ w.r.t. any family U ⊂ P (X).

Definition 1.4. Let φ ∈ O(X, τ) and A, B ⊂ X. Then A is called φ-open if A ⊂ φ(A) and B is called φ-closed if X \ B is φ-open.

X and ∅ are both φ-open and φ-closed sets for any φ ∈ O(X, τ). Let (X, τ) be a topological space, φ ∈ O(X, τ), U ⊂ P (X), x ∈ X. We will use

the following notations.

φO(X) = { U : U ⊂ X, U is φ-open }, 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 1.5. Let φ_{1}, φ_{2} ∈ O(X, τ), X ∈ U ⊂ P (X), A ⊂ X.

a) x ∈ (U-φ_{2}) closure A ⇔ φ_{2}(U) ∩ A ≠ ∅ for each U ∈ U(x).

x ∈ U- closure A ⇔ x ∈ (U-ı) cl A (here ı is the identity operation). x ∈ φ_{1,2} cl A ⇔ x ∈ (φ_{1}O(X)-φ_{2}) cl A.

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

Clearly, for any set A, 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.

For φ_{1}, φ_{2} ∈ O(X, τ), let φ_{1,2}O(X) = {U : U ⊂ X, U is φ_{1,2}-open }.

Many preliminary results and the following theorem can be obtained for topological spaces from [8].

Theorem 1.6. Let φ_{1}, φ_{2} ∈ O(X, τ).

a) φ_{1,2}O(X) is a supratopology on X (U ⊂ P (X) is a supratopology iff ∅ ∈ U, X ∈ U and U is closed under arbitrary unions, [2]).

b) If φ_{2} is regular w.r.t. φ_{1}O(X) then φ_{1,2}O(X) is a topology on X.

c) If φ_{2} ≥ ı or φ_{2} ≥ φ_{1}, and if φ_{2} is regular w.r.t. φ_{1}O(X), then A is closed (open) in the topology φ_{1,2}O(X) iff A = φ_{1,2} cl A (A = φ_{1,2} int A).

Example 1.7. Let φ_{1}, φ_{2} ∈ O(X, τ).

1. If φ_{1} = int, φ_{2} = cl, then φ_{1,2}O(X) is the topology of all Θ-open sets.

2. If φ_{1} = int, φ_{2} = int ∘ cl, then φ_{1,2}O(X) is the topology τ_{s} having a base which is the family of all regular-open sets.

3. If φ_{1} = int ∘ cl ∘ int, φ_{2} = ı, then φ_{1,2}O(X) = τ^{α}, the topology of all α-open sets (A is α-open means that A ⊂ (\overline{A^{o}})^{o}).

2. The W.F.I.P. Property

For φ_{1}, φ_{2} ∈ O(X, τ), if φ_{2} ≥ ı or φ_{2} ≥ φ_{1}, then φ_{1}O(X) ⊂ φ_{2}O(X).

Theorem 2.1. Let (X, τ) be a topological space, X ∈ U ⊂ P (X), A ⊂ P (X), φ_{1}, φ_{2} ∈ O(X, τ). The following are valid:

1. A ⊂ U-cl A for each A ∈ P (X).

2. A ⊂ φ_{1}O(X)-cl A for each A ∈ P (X).

3. If A ⊂ B then U-cl A ⊂ U-cl B.

4. If φ_{2} is W.F.I.P. w.r.t. A, then φ_{2} is W.F.I.P. w.r.t. any subfamily of A.

5. If φ_{2} is W.F.I.P. w.r.t. A, then (U-φ_{2}) cl A ⊂ U-cl A for each A ∈ A.

6. If φ_{2} is W.F.I.P. w.r.t. φ_{1}O(X), then φ_{1,2} cl A ⊂ φ_{1}O(X)-cl A for each A ∈ φ_{1}O(X).

7. If φ_{2} ≥ ı, then U-cl A ⊂ (U-φ_{2}) cl A for each A ∈ P (X).

8. If φ_{2} ≥ ı or φ_{2} ≥ φ_{1}, then φ_{1}O(X)-cl A ⊂ φ_{1,2} cl A for each A ∈ P (X).

9. If φ_{2} ≥ ı or φ_{2} ≥ φ_{1}, and if φ_{2} is W.F.I.P. w.r.t. A then φ_{1} O(X)-cl A = φ_{1,2} cl A for each A ∈ A.

10. If φ_{2} is W.F.I.P. w.r.t. A and φ_{2} ≥ ı then U-cl A = (U-φ_{2}) cl A for each A ∈ A.

Proof. 5) Let φ_{2} be W.F.I.P. w.r.t. A, A ∈ A and x ∈ (U-φ_{2}) cl A.

x ∈ U ∈ U ⇒ φ_{2}(U) ∩ A ≠ ∅
⇒ φ_{2}(U) ∩ A ⊂ φ_{2}(U ∩ A) ≠ ∅
⇒ U ∩ A ≠ ∅.

So x ∈ U-cl A.

8) Let A ∈ P (X) and x ∈ φ_{1}O(X)-cl A.

U ∈ φ_{1}O(X, x) ⇒ U ∩ A ≠ ∅

⇒ φ_{2}(U) ∩ A ≠ ∅ (since U ⊂ φ_{2}(U) or U ⊂ φ_{1}(U) ⊂ φ_{2}(U)).

So x ∈ φ_{1,2} cl A.

For a topological space (X, τ), RO(X), SO(X), P O(X) and τ^{α} will stand for the family of regular open sets, semi-open sets, pre-open sets, α-open sets, respectively.

Example 2.2. Let (X, τ) be a topological space.

1. φ_{2} = cl is W.F.I.P. w.r.t. τ and φ_{2} ≥ ı.

So for any φ_{1} ∈ O(X, τ) and for any U ∈ τ, we have φ_{1,2} cl U = φ_{1}O(X)-cl U.

(a) If we take φ_{1} = int, φ_{1,2} cl A = Θ-cl A, φ_{1}O(X)-cl A = τ-cl A for each A ∈ P (X). So Θ-cl U = \overline{U} for each U ∈ τ.

(b) If we take φ_{1} = cl ∘ int, then for each U ∈ τ, φ_{1,2} cl U = {x : x ∈ V ∈ SO(X) ⇒ \overline{V} ∩ U ≠ ∅\} = Θ-semi-cl U = scl U.

2. Let φ_{1} = int, φ_{2} = int ∘ cl (or φ_{2} = int ∘ cl ∘ int).

φ_{2} is W.F.I.P. w.r.t. φ_{1}O(X) = τ, φ_{2} ≥ φ_{1}. For any A ⊂ X, φ_{1,2} cl A = τ_{s} cl A, φ_{1}O(X)-cl A = τ cl A.

So if U ∈ τ, τ_{s} cl U = τ cl U. We know that if a family A ⊂ P (X) is a supratopology, then

τ_{A} = {U ⊂ X : A ∈ A ⇒ U ∩ A ∈ A}

is a topology on X [17].

If φ ∈ O(X, τ) is monotonous then φO(X) is a supratopology. But the converse is not true.

Example 2.3. Let τ be the usual topology on ℝ and φ: P (ℝ) → P (ℝ) be defined as follows:

φ(A) = \overline{A} if A^{o} = ∅ and φ(A) = A if A^{o} ≠ ∅.

Then φ ∈ O(ℝ, τ), φO(ℝ) = P (ℝ). But φ is not monotonous. For example, for the sets A = (0, 1) ∩ ℚ, B = (0, 1) we have A ⊂ B but φ(A) ⊄ φ(B).

If φO(X) is a supratopology for φ ∈ O(X, τ) and if φ is W.F.I.P. w.r.t. τ then we have τ ⊂ τ_{φO(X)}.

Let (X, τ) be a topological space and let A be a supratopology on X containing τ. If we define the mapping φ: P (X) → P (X) as φ(A) = A-int A (i.e, φ is the interior operation with respect to A) then φ ∈ O(X, τ), φ is monotonous and φO(X) = A. In this case if φ = A-int is W.F.I.P. w.r.t. some subfamily W of P (X) then W ⊂ τ_{A} = τ_{φO(X)}.

For any A ⊂ P (X) containing X, the mapping φ: P (X) → P (X) defined by φ(A) = A-cl A is an operation on (X, τ) for any topology τ on X.

Theorem 2.4.Let A be a supratopology on X and τ_{A}, the topology mentioned above. If we take φ_{2} = A-cl, then φ_{2} is W.F.I.P. w.r.t. τ_{A} and φ_{2} ≥ ı.

Proof. Straightforward.

Corollary 2.5.Let φ ∈ O(X, τ). If φO(X) is a supratopology and if we take φ_{2} = φO(X)-cl, then for any φ_{1} ∈ O(X, τ) and for any U ∈ τ_{φO(X)} we have φ_{1,2}clU = φ_{1}O(X)-cl U.

Example 2.6. Take φ_{1}, φ_{2} ∈ O(X, τ).

1. If we take φ_{1} = cl ∘ int, then φ_{1}O(X) = SO(X) and τ_{SO(X)} = τ^{α}. For φ_{2} = φ_{1}O(X)-cl = scl, we get φ_{1,2} cl U = semi-Θ- cl U = scl U for each α-open set U (so for U ∈ τ, U ∈ τ_{s}, U ∈ τ_{θ}).

2. If we take φ_{1} = int ∘ cl, φ_{2} = φ_{1}O(X)-cl = pcl, we get

φ_{1,2} cl U = { x : x ∈ V ∈ P O(X) ⇒ (pcl V) ∩ U ≠ ∅ }
= φ_{1}O(X)- cl U = pcl U for each U ∈ τ_{P O(X)}.

3. If we take A = SO(X), φ_{2} = SO(X)-cl = scl, then φ_{2} ≥ ı, φ_{2} is W.F.I.P. w.r.t. τ^{α}. If we take φ_{1} = int, we get

φ_{1,2} cl A = { x ∈ X : x ∈ U ∈ τ ⇒ scl U ∩ A ≠ ∅ }
= \{ x ∈ X : x ∈ U ∈ τ ⇒ \overline{U}^{o} ∩ A ≠ ∅ \}
= τ_{s} cl A = φ_{1}O(X)- cl A = τ- cl A

for each A ∈ τ^{α} by using Theorem 2.1(9) and the property scl U = U ∪\overline{U}^{o} = \overline{U}^{o} for U ∈ τ [3].

4. Let φ_{1} = int ∘ cl ∘ int, φ_{2} = scl. Then φ_{2} is W.F.I.P. w.r.t. φ_{1}O(X) = τ^{α}. For each A ∈ τ^{α} = φ_{1}O(X) we have φ_{1,2} cl A = {x ∈ X : x ∈ U ∈ τ^{α} ⇒ (scl U) ∩ A ≠ ∅} = {x ∈ X : x ∈ U ∈ τ^{α} ⇒ \overline{U}^{o} ∩ A ≠ ∅\} = \{x ∈ X : x ∈ U ∈ τ ⇒ \overline{U}^{o} ∩ A ≠ ∅\} = τ_{s}- cl A = φ_{1}O(X)- cl A = τ^{α} cl A.

5. If we take φ_{2} = pcl, then φ_{2} ≥ ı, φ_{2} is W.F.I.P. w.r.t. τ_{P O(X)} (and w.r.t. τ, τ_{s}, τ_{θ}). If we take φ_{1} = int, we get φ_{1,2} cl A = {x ∈ X : x ∈ T ∈ τ ⇒ (pcl T) ∩ A ≠ ∅\} = \{x ∈ X : x ∈ T ∈ τ ⇒ \overline{T^{o}} ∩ A ≠ ∅\} = \{x ∈ X : x ∈ T ∈ τ ⇒ \overline{T} ∩ A ≠ ∅\} = Θ- cl A = φ_{1}O(X)- cl A = τ cl A for each A∈ τ_{P O(X)} (so for A∈ τ, A ∈ τ_{s}, A ∈ τ_{θ}). We used here the property pcl A = A ∪ \overline{A^{o}} for any A ⊂ X [3]. It can be seen from the above examples that we may obtain many equalities

for many different types of closure.

Lemma 2.7. Let φ_{1}, φ_{2} ∈ O(X, τ). If for each U∈ φ_{1}O(X) we have φ_{2}(U) ∈ φ_{1}O(X) and φ_{2}(φ_{2}(U)) ⊂ φ_{2}(U), then for each U ∈ φ_{1}O(X), φ_{2}(U) is φ_{1,2}-open.

The definition of a base of a supratopology, given in [2], is similar to that of a base of a topology.

Theorem 2.8.Take φ_{1}, φ_{2} ∈ O(X, τ). If φ_{2} ≥ φ_{1} or φ_{2} ≥ ı, and if φ_{2}(U) is φ_{1,2}-open for each U ∈ φ_{1}O(X), then the family B = {φ_{2}(U) : U ∈ φ_{1}O(X)} is a base for the supratopology φ_{1,2}O(X).

Proof. Firstly, B ⊂ φ_{1,2}O(X) by hypothesis. Let A be a φ_{1,2}-open set. Then

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

Hence, A = ⋃{φ_{2}(U_{x}) : x ∈ A}.

Example 2.9. Take φ_{1}, φ_{2} ∈ O(X, τ).

1. If φ_{1} = int, φ_{2} = int ∘ cl, then A is φ_{1,2}-open ⇔ A ∈ τ_{s}.

For U ∈ φ_{1}O(X) = τ, φ_{2}(U) = \overline{U}^{o} ∈ τ_{s}. B = \{φ_{2}(U) : U ∈ φ_{1}O(X)\} = \{\overline{U}^{o} : U ∈ τ\} = RO(X). This gives us the known result that RO(X) is a base for τ_{s}.

2. If φ_{1} = cl ∘ int, φ_{2} = scl, then φ_{1}O(X) = SO(X). For U ∈ SO(X), φ_{2}(U) = scl U ∈ SO(X), φ_{2} ≥ ı. Thus A is φ_{1,2}-open iff A is semi-Θ-open. Hence, B = {scl U : U ∈ SO(X)} = SR(X) is a base for the supratopology of semi-Θ-open sets.

3. φ_{1,2}ψ_{1,2}-continuities

We accept throughout that the operations φ_{i} (i = 1, 2, 3, 4), (ψ_{i} (i = 1, 2, 3, 4)) are defined on a topological space (X, τ)((Y, ϑ), resp.), and that f is a function from (X, τ) to (Y, ϑ).

The following definition, Theorem 3.2 and Theorem 3.3 are given for fuzzy topological spaces in [6].

Definition 3.1. A function f: (X, τ) → (Y, ϑ) is called φ_{1,2}ψ_{1,2}-continuous if for each x ∈ X and for each V∈ ψ_{1}O(Y, f(x)), there exists a U ∈ φ_{1}O(X, x) such that f(φ_{2}(U)) ⊂ ψ_{2}(V).

Theorem 3.2.For a function f: (X, τ) → (Y, ϑ) the following are equivalent:

(a) f is φ_{1,2}ψ_{1,2} continuous.

(b) For each V ∈ ψ_{1}O(Y) we have f^{−1}(V) ⊂ φ_{1,2} int f^{−1}(ψ_{2}(V)).

(c) For each x ∈ X and for each V ∈ ψ_{1}O(Y, f(x)), x ∈ φ_{1,2} int f^{−1}(ψ_{2}(V)).

(d) For each A ∈ P (X) we have f(φ_{1,2} cl A) ⊂ ψ_{1,2} cl f(A).

(e) For each B ∈ P (Y) we have φ_{1,2} cl f^{−1}(B) ⊂ f^{−1}(ψ_{1,2} cl(B)).

(f) For each B ∈ P (Y) we have f^{−1}(ψ_{1,2} int B) ⊂ φ_{1,2} int f^{−1}(B).

Theorem 3.3.Let f: (X, τ) → (Y, ϑ). The following are valid:

(a) If f is φ_{1,2}ψ_{1,2}-continuous then the inverse image of each ψ_{1,2}-open set is φ_{1,2}-open.

(b) The inverse image of each ψ_{1,2}-open set is φ_{1,2}-open iff the inverse image of each ψ_{1,2}-closed set is φ_{1,2}-closed.

(c) If φ_{1} ≤ φ_{3}, φ_{2} ≥ φ_{4}, ψ_{1} ≥ ψ_{3}, ψ_{2} ≤ ψ_{4} and f is φ_{1,2}ψ_{1,2}-continuous then f is φ_{3,4}ψ_{3,4}-continuous.

(d) If for each x ∈ X and for each V ∈ ψ_{1}O(Y, f(x)), there exists a φ_{1,2}-open set U containing x such that f(U) ⊂ ψ_{2}(V), then f is φ_{1,2}ψ_{1,2}-continuous.

Theorem 3.4. If ψ̃_{2} is the dual operator of ψ_{2}, then f: (X, τ) → (Y, ϑ) is φ_{1,2}ψ_{1,2}-continuous iff φ_{1,2} cl(f^{−1}(ψ̃_{2}(K))) ⊂ f^{−1}(K) for each ψ_{1}-closed subset K of Y.

Theorem 3.5. If ψ_{1} is the dual operator of ψ_{2}, then f: (X, τ) → (Y, ϑ) is φ_{1,2}ψ_{1,2}-continuous iff φ_{1,2} cl f^{−1}(ψ_{1}(K)) ⊂ f^{−1}(K) for each ψ_{1}-closed set K of Y iff for each ψ_{1}-open set U, f^{−1}(U) ⊂ φ_{1,2} int f^{−1}(ψ_{2}(U)).

Theorem 3.6. If ψ_{2} ≥ ψ_{1} or ψ_{2} ≥ ı, and if ψ_{2}(V) is ψ_{1,2}-open for each V ∈ ψ_{1}O(Y), then the following are equivalent for a function f: (X, τ) → (Y, ϑ).

(a) f is φ_{1,2}ψ_{1,2}-continuous.

(b) The inverse image of each ψ_{1,2}-open set is φ_{1,2}-open.

(c) f^{−1}(ψ_{2}(V)) is φ_{1,2}-open for each V ∈ ψ_{1}O(Y).

Proof. (a) ⇒ (b). Known from Theorem 3.3.

(b) ⇒ (c). Obvious from the hypothesis.

(c) ⇒ (a). Let x ∈ X and V ∈ ψ_{1}O(Y, f(x)). Since ψ_{1} ≤ ψ_{2} or ı ≤ ψ_{2} we have f(x) ∈ V ⊂ ψ_{2}(V) so x ∈ f^{−1}(ψ_{2}(V)) and f^{−1}(ψ_{2}(V)) is φ_{1,2}-open. There exists an U ∈ φ_{1}O(X, x) such that φ_{2}(U) ⊂ f^{−1}(ψ_{2}(V)). Hence f(φ_{2}(V)) ⊂ ψ_{2}(V), so f is φ_{1,2}ψ_{1,2}-continuous.

Theorem 3.7.If ψ_{2} = ı, then a function f: (X, τ) → (Y, ϑ) is φ_{1,2}ψ_{1,2}- continuous iff f^{−1}(V) is φ_{1,2}-open for each V ∈ ψ_{1}O(Y).

Theorem 3.8.If ψ_{1} is the dual of ψ_{2} and ψ_{1}(B) ∈ ψ_{1}O(Y), ψ_{1,2} cl(ψ_{1}(B)) ⊂ ψ_{2}(B) for each B ⊂ Y, then the following are equivalent.

(a) f is φ_{1,2}ψ_{1,2}-continuous.

(b) φ_{1,2} cl(f^{−1}(V)) ⊂ f^{−1}(ψ_{1,2} cl(V)) for each V ∈ ψ_{1}O(Y).

(c) f^{−1}(ψ_{1,2} int(K)) ⊂ φ_{1,2} int f^{−1}(K) for each ψ_{1}-closed set K.

Theorem 3.9. If φ_{2} and ψ_{2} are monotonous then the following are equivalent for a function f: (X, τ) → (Y, ϑ).

(a) f is φ_{1,2}ψ_{1,2}-continuous.

(b) For each x ∈ X and for each N ∈ N (ψ_{1}O(Y), f(x)), there exists an M ∈ N (φ_{1}O(X), x) such that f(φ_{2}(M)) ⊂ ψ_{2}(N).

(c) For each x ∈ X and for each N ∈ N (ψ_{1}O(Y), f(x)), x ∈ φ_{1,2} int f^{−1}(ψ_{2}(N)).

Theorem 3.10 If ψ_{2} is monotonous and φ_{2} = ı, then any function f: (X, τ) → (Y, ϑ) is φ_{1,2}ψ_{1,2}-continuous iff for each x ∈ X and for each N ∈ N (ψ_{1}O(Y), f(x)) we have f^{−1}(ψ_{2}(N)) ∈ N (φ_{1}O(X), x).

Remark 3.11. Clearly if a family B is a base for the supratopology ψ_{1,2}O(Y), then f^{−1}(V) ∈ φ_{1,2}O(X) for each V ∈ ψ_{1,2}O(Y) iff f^{−1}(V) ∈ φ_{1,2}O(X) for each V ∈ B iff f^{−1}(Y \ V) is φ_{1,2}-closed for each V ∈ B.

Theorem 3.12. If φ_{2} ≥ φ_{1} or φ_{2} ≥ ı, and if φ_{2}(U) is φ_{1,2}-open for each U ∈ φ_{1}O(X) then (noting that B = {φ_{2}(U) : U ∈ φ_{1}O(X)} is a base of the supratopology φ_{1,2}O(X) by Theorem 2.8.), the following are equivalent.

(a) f is φ_{1,2}ψ_{1,2}-continuous.

(b) For each x ∈ X and for each V ∈ ψ_{1}O(Y, f(x)), there exists a B ∈ B containing x such that f(B) ⊂ ψ_{2}(V).

(c) For each x ∈ X and for each V ∈ ψ_{1}O(Y, f(x)), there exists a φ_{1,2}-open set W containing x such that f(W) ⊂ ψ_{2}(V).

Proof. (b) ⇒ (c) ⇒ (a). Clear from the hypotheses and Theorem 3.3(d).

(a) ⇒ (b). Let x ∈ X and V ∈ ψ_{1}O(Y, f(x)). Then there exists a U ∈ φ_{1}O(X, x) such that f(φ_{2}(U)) ⊂ ψ_{2}(V) and x ∈ φ_{2}(U). Since φ_{2}(U) ∈ B, we obtain the desired result.

Example 3.13. Let φ_{1} = cl ∘ int, φ_{2} = scl on (X, τ) and ψ_{1} = int, ψ_{2} = ı on (Y, ϑ). Then:

f: (X, τ) → (Y, ϑ) is φ_{1,2}ψ_{1,2}-continuous iff f is strongly Θ-semi-continuous (strong Θ-semi-continuity was defined in [4], and called s-Θ-continuity in [13]).

φ_{2} ≥ ı, φ_{2}(U) = scl U ∈ φ_{1}O(X) = SO(X) and φ_{2}(φ_{2}(U)) = scl(scl U) = scl U = φ_{2}(U) for each U ∈ φ_{1}O(X).

B = {φ_{2}(U) : U ∈ φ_{1}O(X)} = {sclU : U ∈ SO(X)}

= SR(X) = the family of sets which are semi-open and semi-closed. ψ_{2} ≥ ı, ψ_{1,2}O(Y) = ϑ, ψ_{2}(V) is ψ_{1,2}-open for each V ∈ ψ_{1}O(Y), ψ̃_{2} = ı is the

dual of ψ_{2}, ψ_{1,2} cl B = \overline{B} for each B ∈ P (Y) and φ_{1,2} cl A = semi-Θ- cl A [13].

Now, by using Lemma 2.7, Theorems 2.8, 3.2, 3.3(b), 3.4, 3.6, 3.7, 3.9 and 3.12, we get the following result which generalizes Theorem 2.1 in [4].

Theorem 3.14. For any function f: (X, τ) → (Y, ϑ), the following are equivalent:

(a) f is strongly Θ-semi-continuous.

(b) For each x ∈ X and for each V ∈ ϑ(f(x)) we have x ∈ semi-Θ- int f^{−1}(V).

(c) For each B ∈ P (Y) we have f^{−1}(B^{o}) ⊂ semi-Θ- int f^{−1}(B).

(d) Semi-Θ- cl(f^{−1}(K)) ⊂ f^{−1}(K) for each closed subset K of Y.

(e) f^{−1}(V) is semi-Θ-open for each open subset V of Y.

(f) f^{−1}(K) is semi-Θ-closed for each closed subset K of Y.

(g) f(semi-Θ- cl A) ⊂ \overline{f(A}) for each A ⊂ X.

(h) Semi-Θ- cl(f^{−1}(B)) ⊂ f^{−1}(\overline{B}) for each B ⊂ Y.

(i) For each x ∈ X and for each V ∈ ϑ(f(x)), there exists a semi-regular set U containing x such that f(U) ⊂ V.

(j) For each x ∈ X and for each V ∈ ϑ(f(x)), there exists a semi-Θ-open set W such that f(W) ⊂ V.

(k) For each x ∈ X and for each N ∈ N (ϑ, f(x)), there exists an M ∈ N (SO(X), x) such that f(scl M) ⊂ N.

(l) For each x ∈ X and for each N ∈ N (ϑ, f(x)) we have x ∈ semi-θ- int(f^{−1}(N)).

Example 3.15. Let φ_{1} = int, φ_{2} = ı on (X, τ) and ψ_{1} = int, ψ_{2} = cl on (Y, ϑ). Then:

f is φ_{1,2}ψ_{1,2}-continuous iff f is weakly continuous (weak-continuity was defined by Levine [12]).

φ_{2} and ψ_{2} are monotonous, ψ_{1} is the dual of ψ_{2}, ψ_{2} ≥ ı and ψ_{2} is W.F.I.P. w.r.t. ψ_{1}O(Y) = ϑ.

ψ_{1,2} cl V = Θ- cl V = ψ_{1}O(Y)- cl V = ϑ cl V = \overline{V} = ψ_{2}(V) for each V ∈ ψ_{1}O(Y) = ϑ.

For each B ∈ P (Y) we have ψ_{1}(B) = B^{o} ∈ ψ_{1}O(Y) = ϑ and ψ_{1,2} cl(ψ_{1}(B)) = Θ- cl B^{o} = \overline{B^{o}} ⊂ \overline{B} = ψ_{2}(B).

Now, by using Theorems 2.1(9), 3.2, 3.5, 3.8 and 3.9 we obtain the following result.

Theorem 3.16. For any function f: (X, τ) → (Y, ϑ), the following are equivalent:

(a) f is weakly continuous.

(b) For each x ∈ X and for each V ∈ ϑ(f(x)), there exists an U ∈ τ(x) such that f(U) ⊂ Θ- cl V.

(c) f^{−1}(V) ⊂ (f^{−1}(\overline{V}))^{o} for each V ∈ ϑ.

(d) f^{−1}(V) ⊂ (f^{−1}(Θ- cl V))^{o} for each V ∈ ϑ.

(e) For each x ∈ X and for each V ∈ ϑ(f(x)), we have x ∈ (f^{−1}(\overline{V}))^{o}.

(f) For each x ∈ X and for each V ∈ ϑ(f(x)), we have x ∈ (f^{−1}(Θ- cl V))^{o}.

(g) f(\overline{A}) ⊂ Θ- cl(f(A)) for each A ⊂ X.

(h) \overline{f^{−1}(B)} ⊂ f^{−1}(Θ- cl B) for each B ⊂ Y.

(i) f^{−1}(Θ- int B) ⊂ (f^{−1}(B))^{o} for each B ⊂ Y.

(j) \overline{f^{−1}(K^{o})} ⊂ f^{−1}(K) for each closed subset K of Y.

(k) \overline{f^{−1}(V)} ⊂ f^{−1}(\overline{V}) for each V ∈ ϑ.

(l) \overline{f^{−1}(V)} ⊂ f^{−1}(Θ- cl V) for each V ∈ ϑ.

(m) For each x ∈ X and for each N ∈ N (ϑ, f(x)), there exists an M ∈ N (τ, x) s.t. f(M) ⊂ \overline{N}.

(n) For each x ∈ X and for each N ∈ N (ϑ, f(x)) we have x ∈ (f^{−1}(\overline{N}))^{o}.

If we use the properties: Θ- cl V = \overline{V} = ϑ_{s} cl V = ϑ^{α} cl V for each V ∈ ϑ (Examples 2.2 and 2.6), Θ- int K = K^{o} = ϑ_{s} int K = ϑ^{α} int K for each closed subset K of Y,

then we obtain some other characterizations of weak continuity.

Theorem 3.17. The following are equivalent for any function f: (X, τ) → (Y, ϑ):

(a) f is weakly continuous.

(b) For each x ∈ X and for each V ∈ ϑ(f(x)), there exists a U ∈ τ(x) such that f(U) ⊂ ϑ_{s} cl V.

(c) f^{−1}(V) ⊂ (f^{−1}(ϑ_{s}clV))^{o} for each V∈ ϑ.

(d) For each x ∈ X and for each V ∈ ϑ(f(x)), we have x ∈ (f^{−1}(ϑ_{s} cl V))^{o}.

(e) \overline{f^{−1}(Θ- int K)} ⊂ f^{−1}(K) for each closed subset K of Y.

(f) \overline{f^{−1}(ϑ_{s} int K)} ⊂ f^{−1}(K) for each closed subset K of Y.

(g) \overline{f^{−1}(V)} ⊂ f^{−1}(ϑ_{s} cl V) for each v ∈ ϑ.

(h) For each x ∈ X and for each V ∈ ϑ(f(x)), there exists a U ∈ τ(x) such that f(U) ⊂ ϑ^{α} cl V.

(i) f^{−1}(V) ⊂ (f^{−1}(ϑ^{α}clV))^{o} for each V ∈ ϑ.

(j) For each x ∈ X and for each V ∈ ϑ(f(x)), we have x ∈ (f^{−1}(ϑ^{α} cl V))^{o}.

(k) \overline{f^{−1}(ϑ^{α} int K))} ⊂ f^{−1}(K) for each closed subset K of Y.

Example 3.18. Let φ_{1} = int, φ_{2} = ı on (X, τ) and ψ_{1} = int, ψ_{2} = int ∘ cl on (Y, ϑ). Then:

f: (X, τ) → (Y, ϑ) is φ_{1,2}ψ_{1,2}-continuous iff f is almost-continuous (almostcontinuity was defined in [16]);

φ_{1,2} int A = τ- int A = A^{o}, φ_{1,2} cl A = \overline{A} for any A ⊂ X; ψ̃_{2} = cl ∘ int is the dual operation of ψ_{2}, φ_{2}, ψ_{2} are monotonous and ψ_{2} ≥ ψ_{1}; ψ_{2} is W.F.I.P. w.r.t. ψ_{1}O(Y) = ϑ; ψ_{2}(V) = \overline{V}^{o} is ψ_{1,2}-open for each V ∈ ψ_{1}O(Y), B = \{ψ_{2}(V) : V ∈ ϑ\} =

RO(Y) and RO(Y) is a base for the supratopology ψ_{1,2}O(Y) = ϑ_{s};

ψ_{1,2} int B = δ- int B = ϑ_{s}- int B and ψ_{1,2} cl V = δ- cl V = ϑ_{s} cl V for any B ⊂ Y.

For V ∈ ϑ, ψ_{1,2} cl V = δ- cl V = ϑ_{s} cl V = ψ_{1}O(Y)- cl V = ϑ cl V = \overline{V}. Now, by using Theorems 2.1, 2.8, 3.2, 3.3(b), 3.4, 3.6, 3.9, 3.10, and Remark

3.11, we get the following result.

Theorem 3.19. The following are equivalent for any function f: (X, τ) → (Y, ϑ):

(a) f is almost continuous.

(b) f^{−1}(V) ⊂ (f^{−1}(\overline{V}^{o}))^{o} for each V ∈ ϑ.

(c) For each x ∈ X and each V ∈ ϑ containing f(x) we have x ∈ (f^{−1}(\overline{V}^{o}))^{o}.

(d) f(\overline{A}) ⊂ ϑ_{s}- cl f(A) for each A ∈ P (X).

(e) \overline{f^{−1}(B)} ⊂ f^{−1}(ϑ_{s}- cl B) for each B ∈ P (Y).

(f) f^{−1}(ϑ_{s}- int B) ⊂ (f^{−1}(B))^{o} for each B ∈ P (Y).

(g) \overline{f^{−1}(\overline{K^{o}})} ⊂ f^{−1}(K) for each closed subset K of Y.

(h) f^{−1}(V) is open for each V ∈ ϑ_{s}.

(i) f^{−1}(\overline{V}^{o}) is open for each V ∈ ϑ.

(j) f^{−1}(V) is open for each V ∈ RO(Y).

(k) f^{−1}(K) is closed for each regular-closed subset K of Y.

(l) f^{−1}(K) is closed for each ϑ_{s}-closed subset K of Y.

(m) For each x ∈ X and for each N ∈ N (ϑ, f(x)), there exists an M ∈ N (τ, x) s.t. f(M) ⊂ \overline{N}^{o}.

(n) For each x ∈ X and for each N ∈ N (ϑ, f(x)) we have x ∈ (f^{−1}(\overline{N}^{o}))^{o}.

For V ∈ ϑ, \overline{V}^{o} = V ∪\overline{V}^{o} = scl V 3. So, for V ∈ ϑ, \overline{V}^{o} = scl V = Θ-semi- cl V = semi-Θ- cl V (from Examples 2.2 and 2.6).

For any closed subset K of Y, \overline{K^{o}} = semi- int K = Θ-semi- int K = semi-Θ- int K.

Now, by using the above equalities, we obtain some other characterizations of almost continuity.

Theorem 3.20. The following are equivalent for any function f: (X, τ) → (Y, ϑ):

(a) f is almost continuous.

(b) For each x ∈ X and for each V ∈ ϑ(f(x)), there exists a U ∈ τ(x) such that f(U) ⊂ scl V.

(c) For each x ∈ X and for each V ∈ ϑ(f(x)), there exists a U ∈ τ(x) such that f(U) ⊂ Θ-semi- cl V.

(d) For each x ∈ X and for each V ∈ ϑ(f(x)), there exists a U ∈ τ(x) such that f(U) ⊂ semi-Θ- cl V.

(e) f^{−1}(V) ⊂ (f^{−1}(scl V))^{o} for each V ∈ ϑ.

(f) f^{−1}(V) ⊂ (f^{−1}(Θ-semi- cl V))^{o} for each V ∈ ϑ.

(g) f^{−1}(V) ⊂ (f^{−1}(semi-Θ- cl V))^{o} for each V ∈ ϑ.

(h) For each x ∈ X and each V ∈ ϑ containing f(x) we have x ∈ (f^{−1}(scl V))^{o}.

(i) For each x ∈ X and each V ∈ ϑ containing f(x) we have x ∈ (f^{−1}(Θ-semi- cl V))^{o}.

(j) For each x ∈ X and each V ∈ ϑ containing f(x) we have x ∈ (f^{−1}(semi-Θ- cl V))^{o}.

(k) \overline{f^{−1}(semi-intK)} ⊂ f^{−1}(K) for each closed subset K of Y.

(l) \overline{f^{−1}(Θ-semi-intK)} ⊂ f^{−1}(K) for each closed subset K of Y.

(m) \overline{f^{−1}(semi-Θ- int K)} ⊂ f^{−1}(K) for each closed subset K of Y.

(n) f^{−1}(scl V) is open for each V ∈ ϑ.

(p) f^{−1}(Θ-semi- cl V) is open for each V ∈ ϑ.

(q) f^{−1}(semi-Θ- cl V) is open for each V ∈ ϑ.

Example 3.21. Let φ_{1} = int, φ_{2} = ı, φ_{3} = int and φ_{4} = ı be operations on (X, τ), and ψ_{1} = int, ψ_{2} = int ∘ cl, ψ_{3} = int and ψ_{4} = cl. Then

φ_{1} ≤ φ_{3}, φ_{2} ≥ φ_{4}, ψ_{1} ≥ ψ_{3} and ψ_{2} ≤ ψ_{4}. By using Theorem 3.3 and Example 3.15 it is seen that almost-continuity

implies weak-continuity.

Continuing in this manner, we may obtain the definitions of many types of continuity as well as many equivalent conditions.

REFERENCES

[1] M. E. Abd El-Monsef, F. M. Zeyada and A. S. Mashour, Operations on topologies and their applications on some types of coverings, Ann. Soc. Sci. Bruxelles, 97,4 (1983), 155–172.

[2] M. E. Abd El-Monsef, E. F. Lashien, Local discrete extensions of supratopologies, Tamkang J. Math. 21, 1 (1990), 1–6.

[3] D. Andrijević, On the topology generated by pre-open sets, Mat. Vesnik, 39 (1987), 367–376.

[4] S. Jafari, T. Noiri, On strongly Θ-semi-continuous functions, Indian J. Pure Appl. Math. 29, 11 (1998), 1195–1201.

[5] D. S. Janković, On functions with α-closed graphs, Glasnik Matematički, 18, 38 (1983), 141– 148.

[6] A. Kandil, E. E. Kerre, A. A. Nouh, Operations and mappings on fuzzy topological spaces, Ann. Soc. Sci. Bruxelles 105, 4 (1991), 165–168.

[7] S. Kasahara, Operation-compact spaces, Math. Japonica 24, 1 (1979), 97–105.

[8] E. E. Kerre, A. A. Nouh, A. Kandil, Operations on the class of fuzzy sets on a universe endowed with a fuzzy topology, J. Math. Anal. Appl. 180 (1993), 325–341.

[9] J. K. Kohli, A class of mappings containing all continuous and all semiconnected mappings, Proc. Amer. Math. Soc. 72, 1 (1978), 175–181.

[10] J. K. Kohli, A framework including the theories of continuties and certain non-continuous functions, Note di Mat. 10, 1 (1990), 37–45.

[11] J. K. Kohli, Change of topology, characterizations and product theorems for semi-locally Pspaces, Houston J. Math. 17, 3 (1991), 335–349.

[12] N. Levine, A decomposition of continuity in topological spaces, Amer. Math. Monthly 68 (1961), 44–46.

[13] M. N. Mukherjee, C. K. Basu, On semi-Θ-closed sets, semi-Θ-connectedness and some associated mappings, Bull. Cal. Mat. Soc. 83 (1991), 227–238.

[14] M. N. Mukherjee, G. Sengupta, On π-closedness: A unified theory, Anal. Sti. Univ. “A⊂. I. Cuza” Iaşi, 39, s. 1.a, Mat. 3 (1993), 1–10.

[15] G. Sengupta, M. N. Mukherjee, On π-closedness, a unified theory II, Boll. U. M. I. 7, 8-B (1994), 999–1013.

[16] M. K. Singal, A. R. Singal, Almost-continuous mappings, Yokohama Math. J. 16 (1968), 63–73.

[17] T. H. Yalvaç, Relations between new topologies obtained from old ones, Acta Math. Hungar. 64, 3 (1994), 231–235.

(received 29.06.2000, revised 10.09.2004)

Hacettepe University, Mathematics Department, 06532 Beytepe Ankara, Turkey E-mail: hayal@hacettepe.edu.tr