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Discrete Dynamics in Nature and Society
Volume 2013 (2013), Article ID 298032, 6 pages
http://dx.doi.org/10.1155/2013/298032
Research Article

Some Local Properties of Soft Semi-Open Sets

School of Mathematical Sciences, University of Jinan, Jinan 250022, China

Received 4 March 2013; Accepted 31 March 2013

Academic Editor: Shurong Sun

Copyright © 2013 Bin Chen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We introduce some local properties by soft semi-open sets. For example, soft semi-neighborhoods of the soft point, soft semi-first-countable spaces and soft semi-pu-continuous at the soft point are given. Furthermore, we define soft semi-connectedness and prove that a soft topological space is soft semiconnected if and only if both soft semi-open and soft semi-closed sets are only and .

1. Introduction

Soft set theory [1] was firstly introduced by Molodtsov in 1999 as a general mathematical tool for dealing with uncertain, fuzzy, not clearly defined objects. He has shown several applications of this theory in solving many practical problems in economics, engineering, social science, medical science, and so forth, in [1].

In the recent years, papers about soft sets theory and their applications in various fields have been writing increasingly [26]. Shabir and Naz [7] introduced the notion of soft topological spaces which are defined over an initial universe with a fixed set of parameters. The authors introduced the definitions of soft open sets, soft closed sets, soft interior, soft closure, and soft separation axioms. And the authors got some important results for soft separation axioms. The results are valuable for research in this field. Tanay and Kandemir [8] introduced the definition of fuzzy soft topology over a subset of the initial universe set while Jun et al. [9] studied the ideal theory in BCK/BCI-Algebras based on soft sets. Along the line of Shabir and Naz, Chen introduced the notations of soft semi-open sets in soft topological spaces in [10].

In the present study, we introduce some local properties by soft semi-open sets. For example, soft semi-neighborhoods of the soft point, soft semi-first-countable spaces, and soft semi--continuous at the soft point are given. And some of their properties are studied.

2. Preliminaries

Let be an initial universe set and be a collection of all possible parameters with respect to , where parameters are the characteristics or properties of objects in . We will call the universe set of parameters with respect to .

Definition 1 (see [1]). A pair is called a soft set over if and , where is the set of all subsets of .

Definition 2 (see [1]). Let be an initial universe set and be a universe set of parameters. Let and be soft sets over a common universe set and . Then is a subset of , denoted by , if: (i) ; (ii) for all , .
equals , denoted by , if and .

Definition 3 (see [1]). A soft set over is called a null soft set, denoted by , if , .

Definition 4 (see [1]). A soft set over is called an absolute soft set, denoted by , if , .

Definition 5 (see [1]). The union of two soft sets and over a common universe is the soft set , where , and for all ,
We write .

Definition 6 (see [1]). The intersection of two soft sets of and over a common universe is the soft set , where , and for all , . We write .
Now we recall some definitions and results defined and discussed in [6, 7, 10]. Henceforth, let be an initial universe set and be the fixed nonempty set of parameter with respect to unless otherwise specified.

Definition 7. For a soft set over , the relative complement of is denoted by and is defined by , where is a mapping given by for all .

Definition 8. Let be the collection of soft sets over , then is called a soft topology on if satisfies the following axioms.(1) belong to .(2)The union of any number of soft sets in belongs to .(3)The intersection of any two soft sets in belongs to .
The triplet is called a soft topological space over .

Definition 9. Let be a soft space over , then the members of are said to be soft open sets in .

Definition 10. Let be a soft space over . A soft set over is said to be a soft closed set in , if its relative complement belongs to .

Proposition 11. Let be a soft space over . Then one has the following.(1) are soft closed sets over .(2)The intersection of any number of soft closed sets is a soft closed set over .(3)The union of any two soft closed sets is a soft closed set over .

Definition 12. Let be a soft topological space and be a soft set over .(1)The soft interior of is the soft set is soft open and .(2)The soft closure of is the soft set is soft closed and .
By property 2 for soft open sets, is soft open. It is the largest soft open set contained in .
By property 2 for soft closed sets, is soft closed. It is the smallest soft closed set containing .

Proposition 13. Let be a soft topological space and let and be a soft set over . Then(1),(2) implies ,(3),(4) implies .

Definition 14. A soft set in a soft topological space will be termed soft semi-open (written S.S.O) if and only if there exists a soft open set such that .

Definition 15. A soft set in a soft topological space will be termed soft semi-closed (written S.S.C) if its relative complement is soft semi-open, for example, there exists a soft closed set such that .

Definition 16. Let be a soft topological space and be a soft set over .(1)The soft semi-interior of is the soft set is soft semi-open and .(2)The soft semi-closure of is the soft set is soft semi-closed and .

3. Soft Semi-Neighborhoods

Definition 17. A soft set in a soft topological space is said to be a soft semi-neighborhood of the soft point if there is a soft semi-open set s.t. .
The semi-neighborhood system of a soft point which is denoted by is the set of all its semi-neighborhoods.

Proposition 18. The semi-neighborhood system at a soft point in the soft topological space has the following results.(a)If , then one has .(b)If and , then one has .(c)If , then there exists a soft set s.t. for each .

Proof. (a) If , then we have a soft semi-open set s.t. . So we have .
(b) If and . Because , there exists a soft semi-open set s.t. . So we have and we have .
(c) If , then there is a soft semi-open set s.t. . Let and for each . This shows .

Definition 19. Let be a soft topological space and be a soft semi-neighborhood of a soft point . If for every soft semi-neighborhood of , there is a such that , then is said to be a soft semi-neighborhoods base of at .

Proposition 20. Let be a soft topological space and be a soft semi-neighborhood of a soft point . is the soft semi-neighborhoods base of at . Then one has the following.(a)If , then we have .(b)If and , then one has .(c)If , then there exists a soft set s.t. for each .

Proof. These properties are easily verified by referring to the corresponding properties of soft seminbds in Proposition 18.

Definition 21. Let be a soft topological space and be a soft point in . If has a countable soft semi-neighborhoods base, then we say that is soft semi-first-countable at . If each soft point in is soft semi-first-countable, then we say that is a soft semi-first-countable space.

Proposition 22. Let be a soft topological space and be a soft point in . Then is soft semi-first-countable at if and only if there is a countable soft semi-neighborhoods base at such that for each .

Proof. Obvious.
Let be a countable soft semi-neighborhoods base at . For each , put . Then it is easy to see that is a soft semi-neighborhoods base at and for each .

Definition 23. Let be a soft topological space, be a soft set, and be a soft point in . Then is said to be a soft semi-interior point of if there is a soft semi-open set satisfies .

Proposition 24. Let be a soft point in and be a soft semi-open set. Then one has the following results.(a)Each soft point is a soft semi-interior point.(b)For each , one can define as follows:Then is a soft semi-interior point of and for each soft semi-interior point of .(c).

Proof. (a), (c) are obvious by definitions.
(b) For is a soft semi-open set, the soft semi-interior point is the largest soft semi-interior point of by and we have for every soft semi-interior point of .

Proposition 25. Let be a soft topological space, be a soft set. Then is any soft semi-interior point of .

Proof. By the definition of soft semi-interior and Proposition 24.

Definition 26. A soft set in a soft topological space is called a soft semi-neighborhood (briefly: snbd) of the soft set if there is a soft semi-open set s.t. .

Proposition 27. Let be a soft topological space, be a soft set. Then is soft semi-open if and only if for each soft set contained in , is a soft semi-nbd of .

Proof. obvious
Because , then we have a soft semi-open set s.t. . So we have and is soft semi-open.

Definition 28 (see [6]). Let the sequence be a soft sequence in a soft topological space . Then is eventually contained in a soft set if and only if there is an integer such that, if , then . The sequence is frequently contained in if and only if for each integer , there is an integer such that and .

Definition 29. Let the sequence be a soft sequence in a soft topological space , then one says semiconverges to a soft point if it is eventually contained in each semi-nbd of . And the soft point is said to be a semicluster soft point of if the sequence is frequently contained in every semi-nbd of .

Proposition 30. Let be soft semi-first-countable, then one has the following.(a) is soft semi-open for every soft sequence which semiconverges to in is eventually contained in .(b)If is a semicluster soft point of the soft sequence , then one has a subsequence of which semi-converges to .

Proof. (a) Because is soft semi-open, is a semi-nbd of , and semi-converges to . Then we have is eventually contained in .
For each contained in , let be the semi-nbd systems such that for each by Proposition 22. Then is eventally contained in each semi-nbd of that is, semi-converges to . So we have an integer such that, if , . Then is a semi-nbd of , and by Proposition 27, is soft semi-open.
(b) Let be the seminbds such that for each by Proposition 22. For every nonnegative integer , find satisfies and . Then is a subsequence of the sequence . Obviously this subsequence semiconverges to .

4. Soft Semi--Continuous Functions

In this part, we define the soft semi--continuous functions induced by two mapping and on soft topological spaces , and study some properties on them. We use and denote all soft sets on and . and denote all semi-open soft sets on soft topological spaces and .

Definition 31. Let and be two soft topological spaces. Let and be mappings. Let be a function and be a soft point in .(a) is soft semi--continuous at if for any , there is a s.t. .(b) is soft semi--continuous from to if is soft semi--continuous for every soft point of .

Proposition 32. Let and be two soft topological spaces. Let and be mappings. Let be a function and be a soft point in . Then the following statements are equal.(a) is soft semi--continuous at .(b)For any , there is a satisfies .(c)For any , .

Proof. From Definition 31, this is obvious.

Proposition 33. Let and be two soft topological spaces. Let and be mappings. be a function. Then the following statements are equal.(a) is soft semi--continuous.(b)If , one has .(c)For each soft semi-closed set , one has is soft semi-closed in .

Proof. (a) (b) Let and . Next, we will prove that . Because and , we have . Because is soft semi--continuous at , there is a s.t. . So we have and . By Proposition 27, .
(b) (a) Let be a soft point, . Then we can get a soft semi-open set s.t. . By (b) and . And we have . So, we prove that is soft semi--continuous at each soft point.
(b) (c) Let be soft semi-closed in . Then and by (b), . Because , we prove is soft semi-closed in .
(c) (b) Using the same method in (b) (c), we can get the result.

Proposition 34. Let and be two soft topological spaces. Let and be mappings. be a function. If is soft semi-first-countable, then the following statements are equal.(a) is soft semi--continuous.(b)If , the inverse image of any semi-nbd of is a semi-nbd of .(c)If and for every semi-nbd of , there exists a semi-nbd of satisfies .(d)Let the soft sequence semi-converges to , then one has semi-converges to .

Proof. (a) (b) Let be soft semi--continuous. If is a semi-nbd of , then contains a soft semi-open nbd of . Because , we have . Since and is soft semi-open. So, we have is a semi-nbd of .
(b) (c) Let be a soft set and be any semi-nbd of . Then by (b), is a semi-nbd of . So there is a soft semi-open set s.t. . So we have .
(c) (d) Let be a semi-nbd of , then there exists a semi-nbd of s.t. . Since the sequence of soft sets semi-converges to , there is an such that , for the semi-nbd of , . So we have . Thus, semi-converges to .
(d) (a) Let be any semi-open soft set over . We show is semi-open soft set over . Let be any soft point of . Because is semi-soft first-countable, then there exists a countable soft semi-neighborhoods base at such that for each . Because is decreasing, there exists an such that for , . We know each is soft semi-neighborhood of , so is a soft semi-neighborhood of by Proposition 18 (b). This complete the proof.

Proposition 35. Let and be two soft topological spaces. Let and be mappings. be a function. Then the following statements are equal.(a) is soft semi--continuous.(b)If , one has .(c)If , one has .(d)If , one has .

Proof. (a) (b) For is soft semi--continuous, then by Proposition 33 (c), is soft semi-closed containing , and thus , which gives (()).
To prove that (b) (c), we apply (b) to and we obtain the inclusion , which gives .
To prove that (c) (d), we apply (c) to and we obtain the inclusion , which gives .
To complete (d) (a), for every we have , and it follows from (d) that . Thus, we have , that is, is soft semi-open in . Then is soft semi--continuous.

5. Soft Semi-Connectedness

Definition 36. Let be a soft topological space. A soft semiseparation on is a pair and of nonnull soft semi-open sets s.t. , .

Definition 37. A soft topological space is called soft semi-connected space if there is no soft semiseparations on . And if has such soft semiseparations, then is called soft semi-disconnected space.

Theorem 38. Let be a soft topological space. and are semiseparations on . If is a soft semi-connected subspace of , then one has or .

Proof. Because and are soft semi-open sets, then we have and are also soft semi-open sets. Hence and are semiseparations of . And this is a contradiction. So, one of and is and thus or .

Theorem 39. Let be a soft topological space and is a soft semi-connected subspace of . If , then is soft semi-connected.

Proof. Suppose is not soft semi-connected, then there exist nonnull soft semi-open sets and which form a soft semiseparation of . Then by Theorem 38, we have or . Suppose , then we have . So . Hence , a contradiction. Thus, we have is soft semi-connected.

Theorem 40. A soft topological space is soft semi-connected if and only if the both soft semi-open and soft semi-closed soft sets are only and .

Proof. Let the soft topological space be soft semi-connected. If is both soft semi-open and soft semi-closed in which is different from and . So is a soft semi-open set in which is different from and . Then and is a soft semiseparation of . This is a contradiction. So the both soft semi-open and soft semi-closed soft sets are only and .
Let and be a soft semiseparation of . Let and by Definition 36. This proves that is both soft semi-open and soft semi-closed in which is different from and . This is a contradiction. So is soft semi-connected.

Corollary 41. A soft topological space is soft semi-disconnected if and only if there exists a nonnull proper soft subset which is both soft semi-open and semi-closed.

Proof. By Theorem 40.

Theorem 42. Let and be two soft topological spaces. be a soft semi--continuous function. If is soft semi-connected, then is soft semi-connected.

Proof. Suppose that is not soft semi-connected. Then there is a soft semiseparation and of . So we have and . Obviously, and are different from . So and are a soft semiseparation of . This forms a contradiction. Thus, is soft semi-connected.

6. Concluding Remarks

In this paper, we introduce the concept of soft semi-neighborhoods of the soft point, soft semi--continuous at the soft point and soft semi-connectedness. And some of their properties are studied. In the study, soft semi-first-countable space is also given. And in the following papers, soft Frechét space, soft sequential space, soft set-Frechét space and the tightness of a soft point and their connection with soft semi-first-countable space need further study. We hope that the results in this paper will be useful in practical life and nature society.

Acknowledgments

This work described here is supported by the Grants from the National Natural Science Foundation of China (NSFC no. 10971186, 11061004, 71140004, 61070241, 11226265), the Natural Scientific Foundation of Shandong Province (ZR2010AM019, ZR2010FM035, ZR2011AQ015, ZR2010AQ012, and 2012BSB01159). This work is also supported by the international cooperation in training projects for outstanding young teachers of higher institutions of Shandong Province. The author is grateful to the anonymous referees for careful checking of the details and for helpful comments that improved this paper.

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