ABSTRACT

In this paper, we introduce the notion of αδ-US spaces. Also we study the concepts of αδ-convergence, sequentially αδ-compactness, sequentially αδ-continunity and sequentially αδ-sub-continuity and derive some of their properties.

1. Introduction

In 1967, A. Wilansky [1] introduced and studied the concept of spaces. Also, the notion of αδ-closed sets of a topological space is discussed by R. Devi, V. Kokilavani and P. Basker [2,3]. The concept of slightly continuous functions is introduced and investigated by Erdal Ekici et al. [4]. In this paper, we define that a sequence in a space is αδ-converges to a point if is eventually in every αδ-open set containing. Using this concept, we define the αδ-US space, Sequentially-αδ-continuous, Sequentially-Nearly- αδ-continuous, Sequentially-Sub-αδ-continuous and Sequentially-αδO-compact of a topological space.

2. Preliminaries

Throughout this paper, spaces X and Y always mean topological spaces. Let X be a topological space and A, a subset of X. The closure of A and the interior of A are denoted by and, respectively. A subset A is said to be regular open (resp. regular closed) if (resp., the δ-interior [5] of a subset A of X is the union of all regular open sets of X contained in A and is denoted by. The subset A is called δ-open if, i.e., a set is δ-open if it is the union of regular open sets. The complement of a δ-open set is called δ-closed.

Alternatively, a set is called δ-closed if, where

. The family of all δ-open (resp. δ-closed) sets in is denoted by (resp.). A subset of

is called α-open [6] if and the complement of a α-open are called α-closed. The intersection of all α-closed sets containing A is called the α-closure of A and is denoted by, Dually, α-interior of A is defined to be the union of all α-open sets contained in A and is denoted by.

We recall the following definition used in sequel.

Definition 2.1. A subset of a space X is said to be

(a) An α-generalized closed [7] (αg-closed) set if whenever and is α-open in

(b) An αδ-closed [8] set if whenever and is αg-open in.

The complement of a αδ-closed set is said to be The intersection of all αδ-closed sets of X containing A is called αδ-closure of A and is denoted by. The union of all αδ-open sets of X contained in A is called αδ-interior of A and is denoted by.

3. αδ-US Spaces

Definition 3.1. A sequence in a space, αδ-converges to a point if is eventually in every αδ-open set containing.

Definition 3.2. A space is said to be αδ-US if every sequence in, αδ-converges to a point of.

Definition 3.3. A space is said to be

(a) if each pair of distinct points and in there exists an αδ-open set in such that and and a αδ-open set in such that and.

(b) if for each pair of distinct points and in there exists an αδ-open sets and such that and,.

Theorem 3.4. Every αδ-US-space is.

Proof. Let be an αδ-US-space and be two distinct points of. Consider the sequence, where for any. Clearly αδ-converges to. Since and is αδ-US, does not αδ-converges to, i.e., there exists an αδ-open set containing but not. Similarly, we obtain an αδ-open set containing but not. Thus, is.

Theorem 3.5. Every -space is αδ-US.

Proof. Let be a space and a sequence in. Assume thatαδ-converges to two distinct points and. Then is eventually in every then is eventually in two disjoint αδ-open sets. This is a contradiction. Therefore, is αδ-US.

Definition 3.6. A subset A of a space is said to be

(a) Sequentially αδ-closed if every sequence in A αδ-converges to a point in A(b) Sequentially αδO-compact if every sequence in A has a subsequence which αδ-converges to a point in A.

Theorem 3.7. A space is αδ-US if and only if the diagonal set Δ is a sequentially αδ-closed subset of the product space.

Proof. Suppose that is an αδ-US space and

is a sequence in the diagonal Δ. It follows that is a sequence in. Since is αδ-US, the sequenceαδ-converges to which clearly belongs to Δ. Therefore, Δ is a sequentially subset of. Conversely, suppose that the diagonal Δ is a sequentially αδ-closed subset of. Assume that a sequence is αδ-converging to x and. Then it follows thatαδ-converges to. By hypothesis, since Δ is sequentially αδ- closed, we have. Thus. Therefore, is αδ-US.

Theorem 3.8. If a space is αδ-US and a subset M of X is sequentially -compact, then M is sequentially αδ-closed.

Proof. Assume that is any sequence in which αδ-converges to a point. Since M is sequentially αδO-compact, there exists a subsequence ofαδ-converges to. Since is αδ-US, we have. This shows that M is sequentially αδ-closed.

Theorem 3.9. The product space of an arbitrary family of αδ-US topological space is an αδ-US topological space.

Proof. Let be a family of αδ-US topological spaces with the index set Δ. The product space of is denoted by. Let be a sequence in. Suppose that

αδ-converges to two distinct points x and y in. Then there exists a such that

. Then is a sequence in.

Let be any αδ-open in containing.

Then is a αδ-open set of

containing x. Therefore, is eventually in. Thus is eventually in and it αδ-converges to. Similarly, the sequenceαδ- converges to. This is a contradiction as is a

αδ-US space. Therefore, the product space is

αδ-US.

4. Sequentially αδO-Compact Preserving Functions

Definition 4.1. A function is said to be

(a) Sequentially-αδ-continuous at if the sequenceαδ-converges to whenever a sequenceαδ-converges to. If is sequentially αδ-continuous at each, then it is said to be sequentially αδ-continuous.

(b) Sequentially-Nearly-αδ-continuous, if for each sequence in that αδ-converges to, there exists subsequence of such that the sequenceαδ-converges to.

(c) Sequentially-Sub-αδ-continuous if for each point and each sequence in αδ-converging tothere exists a subsequence of and a point

such that the sequenceαδ-converges to.

(d) Sequentially, αδO-compact preserving if the image of every sequentially αδO-compact set of is a sequentially αδO-compact subset of.

Theorem 4.2. Let and be two sequentially αδ-continuous functions. If is αδ-US, then the set is sequentially αδ-closed.

Proof. Suppose that is αδ-US and is any sequence in E that -converges to. Since and are sequentially αδ-continuous functions, the sequence (respectively,) converges to (respectively,). Since for each and is αδ-US, and hence. This shows that is sequentially αδ- closed.

Lemma 4.3. Every function is sequentially sub αδ-US αδ-US continuous if is sequentially αδO-compact.

Proof. Let be a sequence in that αδ-US converges to. It follows that is a sequence in. Since is sequentially αδO-compactthere exists a subsequence of that

αδ-converges to a point. Therefore is sequentially sub αδ-continuous.

Theorem 4.4. Every sequentially nearly αδ-continuous function is sequentially αδO-compact preserving.

Proof. Let be a sequentially nearly αδ- continuous function and be any sequentially αδOcompact subset of. We will show that is a sequentially αδO-compact subset of. So, assume that is any sequence in. Then for each, there exists a point such that. Now is sequentially αδO-compact, so there exists a subsequence of that αδ-converges to a point. Since is sequentially nearly αδ-continuous, there exists a subsequence

of such thatαδ-converges to. Therefore, there exists a subsequence of that αδ-converges to. This implies that is a sequentially αδO-compact set of.

Theorem 4.5. Every sequentially αδO-compact preserving function is sequentially sub-αδ-continuous.

Proof. Suppose that is a sequentially -compact preserving function. Let be any point of and a sequence that αδ-converges to. We denote the set by and put. Sinceαδ-converges to, is sequentially αδO-compact. By hypothesis, is sequentially αδO-compact subset of. Now in

there exists a subsequence of that

αδ-converges to a point. This implies that sequentially sub-αδ-continuous.

Theorem 4.6. A function is sequentially -compact preserving if and only if

is sequentially sub-αδ-continuous for each sequentially αδO-compact set of.

Proof. Necessity: Suppose that is a sequentially αδO-compact preserving function. Then is sequentially αδO-compact in for each sequentially αδO-compact subset of. Therefore, by Theorem 3.5 is sequentially sub-αδ-continuous.

Sufficiency: Let be any sequentially αδO-compact set of. We will show that is sequentially αδO-compact subset of. Let be any sequence in. Then for each, there exists a point such that. Since is a sequence in the sequentially αδO-compact set there exists a subsequence of that αδ-converges to a point in. By hypothesis

is sequentially sub-αδ-continuous, hence there exists a subsequence of that αδ-converges to y f(M). This implies that f(M) is sequentially αδO-compact in.

Corollary 4.7. If a function is sequentially sub-αδ-continuous and is sequentially αδ-closed in for each sequentially αδO-compact set M of, then f is sequentially αδO-compact preserving.

Proof. It will be sufficient to show that

is sequentially sub-αδ-continuous for each sequentially αδO-compact set of and by Lemma 3.3. We have already done. So, let be any sequence in that αδ-converges to a point. Then, since is sequentially sub-αδ-continuous there exists a subsequence of and a point such thatαδ-converges to y.

Since is a sequence in the sequentially αδclosed set of, we obtain. This implies that is sequentially sub αδ-continuous.

5. Slightly αδ-Continuous Functions

Definition 5.1. A function is said to be slightly αδ-continuous if for each and for each

, there exists such that, where is the family of clopen sets containing f(x) in a space.

Definition 5.2. Let be a directed set net in is said to be αδ-convergent to a point if is eventually in each .

Theorem 5.3. For a function, the following are equivalent:

(a) is slightly αδ-continuous.

(b) for each.

(c) is αδ-cl-open for each CO(Y).

(d) for each and for each net in.

Proof.. Let and let then. Since is slightly αδ-continuous, there is a such that

. Thus, that is

is a union of αδ-open sets. Hence .

. Let, then.

By hypothesis.

Thus is αδ-closed.

. Let be a net inαδ-converging to and let. There is thus a

such that. There is thus a such that implies since

is αδ-convergent to. Thus

for all. Thus is αδ- convergent to.

Suppose that is not slightly αδ-continuous at a point, then there exists a

such that does not contained in for each. So and thus for each, since is directed by set inclusion, there exists a selection function from into for each. Thus is a net in αδ-converging to. Since and so, for each,

is not eventually in

, which is a contradiction. Hence holds.

Theorem 5.4. If is slightly αδ-continuous and is slightly continuous, then their composition is slightly αδ-continuous.

Proof. Let, then. Since is slightly αδ-continuous,

. Thus is Slightly αδ-continuous.

Theorem 5.5. The following are equivalent for a function:

(a) is slightly αδ-continuous(b) for each and for eachthere exists αδ-cl-open set such that(c) for each closed set of, is αδ- closed(d) for each and

(e) for each.

Proof. Let and

by Theorem 4.3. is clopen.

Put, then and.

is obvious.

since is the smallest αδclosed set containing, hence by, we have.

for each,

. Hence

.

Let. then, by, we have

, since every closed set is αδ-closed, thus

is closed and thus αδ-closed, thus and is slightly αδ-continuous.

Theorem 5.6. If is a slightly αδ-continuous injection and is clopen, then is .

Proof. Suppose that is clopen. For any distinct points and in, there exist such that and . Since is slightly αδ-continuous,

and are αδ-open subsets of such that and . This shows that is.

Theorem 5.7. If is a slightly αδ-continuous surjection and is clopen, then is .

Proof. For any pair of distinct points and in, there exist disjoint clopen sets U and in such that and. Since f is slightly αδ-continuous, and are αδ-open in containing and respectively. Therefore

because. This shows that is.

Definition 5.8. A space is called αδ-regular if for each αδ-closed set and each point, there exist disjoint open sets and such that and .

Definition 5.9. A space is said to be αδ-normal if for every pair of disjoint αδ-closed subsets and of, there exist disjoint open sets and such that and.

Theorem 5.10. If f is slightly αδ-continuous injective open function from an αδ-regular space onto a space then is clopen regular.

Proof. Let F be clopen set in and be, take. Since f is slightly αδ-continuous, is a αδ-closed set, take, we have. Since is αδ-regular, there exist disjoint open sets and such that and. We obtain that and such that f(U) and f(V) are disjoint open sets. This shows that is clopen regular.

Theorem 5.11. If is slightly αδ-continuous injective open function from a αδ-normal space onto a space, then is cl-open normal.

Proof. Let and be disjoint cl-open subsets of Since is slightly αδ-continuous, and

are αδ-closed sets. Take and

. We have. Since is αδ- regular, there exist disjoint open sets A and B such that and. We obtain that and such that and are disjoint open sets. Thus, Y is clopen normal.

REFERENCES

NOTES

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