ABSTRACT

Received October 23, 2013; revised November 23, 2013; accepted November 30, 2013

In this paper, we study the concept of soft sets which is introduced by Molodtsov [5]. We give the definition of -semi open soft (resp. -pre open soft, --open soft, --open soft, -semi open soft, -pre open soft, --open soft, --open soft) set via two soft topologies. Also we introduce -regular open soft and ESDC on two soft topologies. The aim of this paper is to investigate properties of some mixed soft operations and characterizations of ESDC. Finally, we study -ESDC soft topologies.

Keywords:

-Semi Open Soft; -Pre Open Soft; --Open Soft; --Open Soft; -Semi Open Soft; -Pre Open Soft; --Open Soft; --Open Soft; -Regular Open Soft; ESDC

1. Introduction

Some set theories can be dealt with unclear concepts, for example theory of rough sets and theory of fuzzy sets. Unfortunately, these theories are not sufficient to deal with some difficulties and encounter some problems. In 2009, Ali, Feng, Liu, Min and Shabir [1] investigated several operations on soft sets and introduced some new notions such as the restricted intersection, the restricted union etc. In 2011, Hussain and Ahmad [2] researched some properties of soft topological space. Kandil et al. [3] had defined semi-open (pre-open, -open, - open) soft set and semi (pre, ,)-soft continuity via these soft sets. In 2003, Maji, Biswas and Roy [4] introduced equality of two soft sets, subset of a soft set, null soft set, absolute soft set etc. In 1999, Molodtsov [5] introduced the soft theory as a general mathematical tool for dealing with these problems. He accomplished very significant applications of soft set theory such as solving some complications in economics, social science, medical science, engineering etc. In 2011, Shabir and Naz [6] defined and studied some notions such as soft topological space, soft interior, soft closure etc. In 2012, Zorlutuna et al. [7] introduced the concepts of soft interior point, soft interior, soft neighborhood, soft continuity and soft compactness. Later there has been an extensive study on the applications of soft set theory. Many people have studided soft theory and investigated some properties of this theory.

The aim of the present paper is to introduce and study notions of -semi open soft (resp. -pre

open soft, --open soft, --open soft, -semi open soft, -pre-open soft,

--open soft, --open soft) set via two soft topologies. For this purpose, we consider two soft topologies and over. Also we define -regular open soft and ESDC on two soft topologies. Furthermore, we investigate some properties of some mixed soft operations and some characterizations of ESDC. Finally, we show -ESDC soft topologies.

2. Preliminaries

Soft sets and Soft Topology

Definition 2.1 [1]. The complement of a soft set (F,A) is defined as, where , for all.

Theorem 2.1 [2]. Let be a soft topological space over, and are soft sets over. Then

1) and.

2).

3) is a closed set if and only if.

4).

5) implies.

6).

7).

Theorem 2.2 [2]. Let be a soft topological space over and and are soft sets over. Then

1) and.

2).

3).

4) is a soft open set if and only if.

5) implies.

6).

7).

Theorem 2.3 [2]. Let be a soft set of soft topological space over. Then

1).

2).

3).

Definition 2.2 [3]. Let be a soft topological space and. Then is said to be

1) pre-open soft set if;

2) semi-open soft set if;

3) -open soft set if;

4) -open soft set if.

Definition 2.3 [4]. Let and be two soft sets over a common universe X. Then is said to be a soft subset of if and, for all. This relation is denoted by.

Definition 2.4 [4]. A soft set (F,A) over X is said to be a null soft set if, for all. This is denoted by.

Definition 2.5 [4]. A soft set (F,A) over X is said to be an absolute soft set if, for all. This denoted by.

Definition 2.6 [4]. The union of two soft sets and over the common universe is the soft set, where and if or if or if for all.

Definition 2.7 [4]. The intersection of two soft sets and over the common universe is the soft set, where and for all,.

Definition 2.8 [5]. A pair, where is mapping from to, is called a soft set over. The family of all soft sets on is denoted by.

is said to be soft equal to if and. This relation is denoted

by.

Definition 2.9 [6]. Let be the collection of soft sets over X. Then is said to be a soft topology on X if

1);

2) the intersection of any two soft sets in belongs to;

3) the union of any number of soft sets in belongs to.

The triple is called a soft topological space over X. The members of are said to be soft open sets or soft open sets in X. A soft set over X is said to be soft closed in X if its complement belongs to. The set of all soft open sets over denoted by or and the set of all soft closed sets denoted by or.

Definition 2.10 [6]. The difference of two soft sets (F,A) and (G,A) is defined by,

where, for all.

Definition 2.11 [6]. Let be a soft topological space and. The soft closure of, denoted by is the intersection of all closed soft super sets of.

Definition 2.12 [7]. Let be a soft topological space and. The soft interior of, denoted by is the union of all open soft subsets of.

3. Some Properties of Some Mixed Soft Operations

In this section we investigated some properties of some mixed operations such as -semi open soft, -pre open soft. Also we will write for, respectively.

Definition 3.1. Let be an initial universe and be a set of parameters. Let and be two soft topologies on. Then is said to be

1) -semi open soft if;

2) -pre open soft if;

3) --open soft if;

4) --open soft if.

The complement of -semi open (resp. -pre open, --open, --open) soft set is called -semi closed (resp. -pre closed, --closed, --closed) soft (See Figure 1).

Definition 3.2. Let be an initial universe and be a set of parameters. Let and be two soft topologies on. Then is said to be

1) -semi open soft if;

2) -pre open soft if;

3) --open soft if;

-soft open --open soft -semi open soft

4) --open soft if.

The complement of -semi open (resp. -pre open, --open, --open)

soft set is called -semi closed (resp. -pre closed, --closed, --closed)

soft (See Figure 2).

Theorem 3.1. Let be an initial universe and be a set of parameters. Let and be two soft topologies on. Then

1) If and,.

2) If and,.

Proof. It is seen from Definition.

Theorem 3.2. Let be an initial universe and be a set of parameters. Let and be two soft topologies on. Then

1) is a -semi open soft set if and only if.

2) is a -semi open soft set if and only if.

Proof. 1) Necessity. Let be a -semi open soft set. Since, we

have. Also. Hence.

Sufficieny. Let. Therefore and is a -semi open soft.

2) By a similar way.

Theorem 3.3. Let be an initial universe and be a set of parameters. Let and be two soft topologies on. Then

1) If is a -soft open set and is a -pre open soft set, is a - pre open soft.

2) If is a -soft open set and is a -pre open soft set, is a -pre open soft.

Proof. (1). Let be -soft open and be -pre open soft set. Then

from Theorem Hence is a -pre open soft.

(2). By a similar way.

Theorem 3.4. Let be an initial universe and be a set of parameters. Let and be two soft topologies on. Then

1) If either is a -semi open soft or is a -semi open soft set,

.

2) If either is a -semi open soft or is a -semi open soft set,

.

Proof. 1) Let. We have

.

-soft open --open soft -semi open soft

We assume that is a -semi open soft set. Then from Theorem 3.2. So

from Theorem 3.1. Hence we have.

2) By a similar way.

Theorem 3.5. Let be an initial universe and be a set of parameters. Let and be two soft topologies on. Then

1) If is a -soft open set and is a -semi open soft set, is a -semi open soft.

2) If is a -soft open set and is a -semi open soft set, is a -semi open soft.

Proof. 1) Let be -open soft and be -semi open soft set. Then

from Theorem Therefore is a -semi open soft.

2) By a similar way.

Theorem 3.6. Let be an initial universe and be a set of parameters. Let and be two soft topologies on. Let, then

1) is --open soft set if and only if there exists a -soft open set such that

.

2) If is --open soft set and, then is --open soft.

3) is --open soft set if and only if there exists a -soft open set such that

.

4) If is --open soft set and, then is --open soft.

Proof. 1) Necessity. Let. Then

.

Sufficiency. Let is -soft open set and. Then

. Hence. Thus

is --open soft.

2) Let is --open soft set and. Hence

.

Thus is --open soft.

3)-4) By a similar way.

Theorem 3.7. Let be an initial universe and be a set of parameters. Let and be two soft topologies on. Then

1) If is --open soft and is --open soft, then is --open soft.

2) If is --open soft and is --open soft, then

is --open soft.

Proof. 1) Let is --open soft and is --open soft. Then

from Theorem Thus is --open soft.

2) By a similar way.

Proposition 3.1. Let be an initial universe and be a set of parameters. Let and be two soft topologies on. If, the following statements hold:

1).

2).

Proof. It is obvious from Definition 2.1., 2.11. and.

Theorem 3.8. Let be an initial universe and be a set of parameters. Let and be two soft topologies on. Then

1) is a -pre closed soft set if and only if.

2) is a -pre closed soft set if and only if.

Proof. 1) Necessity. Let be -pre closed soft set. Then is a -pre open soft set, that is

.

Thus.

Sufficiency. Let, then.

Hence is a -pre open soft set. Therefore is a -pre closed soft set.

2) By a similar way.

Theorem 3.9. Let be an initial universe and be a set of parameters. Let and be two soft topologies on. Then

1) is a --closed soft set if and only if.

2) is a --closed soft set if and only if.

Proof. 1) Necessity. Let is a --closed soft set. Then is a --open

soft, that is. Therefore,

.

Sufficiency. Let, then

.

Hence is a --open soft set. Therefore is a --closed soft set.

2) By a similar way.

Theorem 3.10. Let be an initial universe and be a set of parameters. Let and be two soft topologies on. Then

1) is a -semi closed soft set if and only if.

2) is a -semi closed soft set if and only if.

Proof. 1) Necessity. Let is a -semi closed soft set. Then is a -semi

open soft set, that is. Therefore,

.

Sufficiency. Let, then.

Hence is a -semi open soft set. Therefore is a -semi closed soft set.

2) By a similar way.

Theorem 3.11. Let be an initial universe and be a set of parameters. Let and be two soft topologies on. Then

1) If is a --open soft set and is a -semi open soft set,

is a -semi open soft.

2) If is a --open soft set and is a -semi open soft set,

is a -semi open soft.

Proof. 1) Let is a --open soft and is a -semi open soft. Then

.

Therefore is a -semi open soft.

2) By a similar way.

Theorem 3.12. Let be an initial universe and be a set of parameters. Let and be two soft topologies on. Then

1) is a --closed soft set if and only if.

2) is a --closed soft set if and only if.

Proof. 1) Necessity. Let is a --closed set. Then is a --open

soft set, that is. There-

fore,.

Sufficiency. Let. Then

.

Hence is a --open soft set. Therefore, is a --closed soft.

2) By a similar way.

4. Extremally Soft Disconnectedness on Two Soft Topologies

In this section we introduced extremally soft disconnectedness (briefly, ESDC) via two soft topological spaces over and investigated some characterizations of ESDC.

Definition 4.1. Let be an initial universe and be a set of parameters. Let and be two soft topologies on. is said to be -ESDC if implies that.

Definition 4.2. Let be an initial universe and be a set of parameters. Let and be two soft topologies on. is said to be -regular open soft set if.

Theorem 4.1. Let be an initial universe and be a set of parameters. Let and be two soft topologies on. Then the following statements are equivalent:

1) is -ESDC.

2) If,.

3) If,.

4) Every -semi open soft set is -pre open soft.

5) If is a --open soft set,.

6) Every --open soft set is -pre open soft.

7) Every -semi open soft set is --open soft.

8) If is a -regular open soft set,.

Proof. 1) Þ 2) Let. Then, that is. From

1),. Hence. Therefore,.

2) Þ 3) Let, Then is t₁-soft closed and from 2)

is -soft closed. Therefore, is -soft open and since

, we have.

3) Þ 4) Let is a -semi open soft set. Then

from 3). Hence, is a -pre open soft.

4) Þ 5) Let is a --open soft set, that is. Then

so that is a -semi open soft, and from 4) it is -

pre open soft.. Hence,.

5) Þ 6) Let is a --open soft set. From 5), and.

Hence,. Therefore, is a -pre open soft.

6) Þ 7) Let is a -semi open soft set, then is a --open soft. From 6), is a -pre open soft. Hence, is a --open soft.

7) Þ 1) Let. Then, so that

and is -semi open soft. From 7), is --open soft. Hence

and so that.

1) Þ 8) Let is -regular open soft. Then and from 1).

Therefore,.

8) Þ 1) Let. Since, the soft set

is -regular open soft. From 8),. Since,

.

Hence, we have so that.

Theorem 4.2. Let be an initial universe and be a set of parameters. Let and be two soft topologies on. Then the following statements are equivalent:

1) is -ESDC.

2) If is -semi open soft set,.

3) If is -pre open soft set,.

4) If is -regular open soft set,.

Proof. 1) Þ 2) Every -semi open soft set is --open soft so that from Theorem 4.1.

2) Þ 4). Every -regular open soft set is -semi open soft set since

.

From 2),.

1) Þ 3) Since every -pre open soft set is --open soft, it is obvious from Theorem 4.1.

3) Þ 4) Since every -regular open soft set is -pre open soft,.

4) Þ 1) If, since and we have

is -regular open soft. Also we obtain from 4) and

by so that

,

.

Therefore.

Lemma 4.1. implies and for.

Proof. Obvious.

Theorem 4.3. Let be an initial universe and be a set of parameters. Let, be two soft topologies on and. Then the following statements are equivalent:

1) is -ESDC.

2) If and,.

3) If, and,.

4) If, and,.

Proof. 1) Þ 2) Let and. From (1),. Then

.

2) Þ 3) Let, and. From (2),

.

3) Þ 4) Let, and. From (3), ,

and,. Hence

.

4) Þ 3) Let, and. From (4),

.

Then we have since. Therefore,

.

Hence.

3) Þ 1) Let. Then and. From (3),

.

Thus. Therefore,.

5. t₂-ESDC Soft Topologies t₁

The family of all semi-open (resp. pre-open, -open, -open) soft sets is denoted by (resp., ,). Also the family of all -semi open (resp. -pre open, --open, --open) soft sets is denoted by (resp., ,.

Theorem 5.1. If and, and implies .

Proof. Let. As, from Lemma 4.1.

is -soft closed for since

.

Hence implies and

.

Therefore, we obtain.

Theorem 5.2. If and, and implies

.

Proof. Let. As, from Lemma 4.1.

is -soft closed for since

.

Hence, implies

.

Therefore, we obtain.

6. Conclusion

We give the definition of -semi open soft (resp. -pre open soft, --open soft, -

-open soft, -semi open soft, -pre open soft, --open soft, --open soft)

set via two soft topologies. Also we introduce -regular open soft and ESDC on two soft topologies. Some properties of some mixed soft operations and characterizations of ESDC are investigated. These properties which are studied are very important for studying anymore.

References