Journal of Applied Mathematics

Volume 2012 (2012), Article ID 508580, 5 pages

doi:10.1155/2012/508580

Research Article

A Note on Some Generalized Closed Sets in Bitopological Spaces Associated to Digraphs

K. Kannan

Department of Mathematics, Srinivasa Ramanujan Centre, SASTRA University, Kumbakonam 612 001, India

Received 2 March 2012; Accepted 29 June 2012

Academic Editor: Livija Cveticanin

Copyright © 2012 K. Kannan. 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.

Many investigations are undergoing of the relationship between topological spaces and graph theory. The aim of this short communication is to study the nature and properties of some generalized closed sets in the bitopological spaces associated to the digraph. In particular, some relations between generalized closed sets in the bitopological spaces associated to the digraph are characterized.

1. Introduction

Concerning the applications of bitopological spaces, there are many approaches to the sets equipped with two topologies of which one may occasionally be finer than the other in analysis, potential theory, directed graphs, and general topology. Lukeš [1] formulated certain new methods to be used in discussing fine topologies, especially in analysis and potential theory in 1977 and one of the properties introduced by him is Lusin-Menchoff property of the fine topologies. This is the initiative to the study of various problems in analysis and potential theory with bitopological spaces.

Brelot [2] compared the notion of a regular point of a set with that of a stable point of a compact set for an analogous Dirichlet problem and thus arrived at a general notion of thinness in classical potential theory.

Bhargava and Ahlborn [3] investigated certain tieups between the theory of directed graphs and point set topology. They obtained several theorems relating connectedness and accessibility properties of a directed graph to the properties of the topology associated to that digraph. Further, they investigated these topologies in terms of closure, kernal, and core operators. This work extended to ceriatn aspects of work done by Bhargava in [4].

Evans et al. [5] proved that there is a one-to-one correspondence between the labelled topologies on points and labelled transitive digraph with vertices. Anderson and Chartrand [6] investigated the lattice graph of the topologies to the transitive digraphs. In particular, they characterized those transitive digraphs whose topologies have isomorphic lattice graphs.

In theoretical development of bitopological spaces [7], several generalized closed sets have been introduced already. Fukutake [8] defined one kind of semiopen sets in bitopological spaces and studied their properties in 1989. Also, he introduced generalized closed sets and pairwise generalized closure operator [9] in bitopological spaces in 1986. A set of a bitopological space is -generalized closed set (briefly - closed) [10] if - whenever and is -open in , and . Also, he defined a new closure operator and strongly pairwise -space. Further study on semiopen sets had been made by Bose [11] and Maheshwari and Prasad [12].

Semi generalized closed sets and generalized semiclosed sets are extended to bitopological settings by Khedr and Al-saadi [13]. They proved that the union of two - closed sets need not be - closed. This is an unexpected result. Also, they defined that the -semi generalized closure of a subset of a space is the intersection of all - closed sets containing and is denoted by -. Rao and Mariasingam [14] defined and studied regular generalized closed sets in bitopological settings. Rao and Kannan [15] introduced semi star generalized closed sets in bitopological spaces in the year 2005. -semi star generalized closed sets [16], regular generalized star star closed sets [17], semi star generalized closed sets [18], and the survey on Levine’s generalized closed sets [19] had been studied in bitopological spaces in 2010, 2011, 2012, 2012, respectively.

The aim of this short communication is to study the nature and properties of some generalized closed sets in the bitopological spaces associated to the digraph. In particular, some relations between generalized closed sets in the bitopological spaces associated to the digraph are characterized.

2. Preliminaries

A digraph is an ordered pair , where is a set and is a binary relation on . A topology may be determined on a set by suitably defining subsets of to be open with respect to the digraph . A set of the digraph is open if there does not exist an edge from to . In other words, a set of the digraph is open if and imply that . A set of the digraph is closed if is open. Consequently, a set of the digraph is closed if there does not exist an edge from to . Equivalently, a set of the digraph is closed if and imply that . Thus, each digraph determines a unique topological space , where ofis open}. Moreover, has completely additive closure. That is, the intersection of any number of open sets is open.

For example, consider the following digraph , where .

Then the topology associated to the above digraph is .

Consequently, and there does not exist an edge from to in forms the topology on and it is denoted by . Hence, we have a unique topological space . Thus, the topology associated to the digraph is .

Now, we are comfortable to define the bitopological space with the help of these two unique topologies associated to the digraph , where are the right and left associated topologies. Also, the topology is called the dual topology to and vise versa so that for every set , the set -cl is the least -open set containing and the set -cl is the least -open set containing . For any set of the digraph , the closure of with respect to is defined by - is accessible from for some. In digraph, -, since is the only point accessible from . Also, -.

To retain the standard notation in the recent trend, will denote the bitopological space . A set is semiopen [20] in a topological space if and the complements of semiopen sets are called semiclosed sets. - and - represent the semiclosure and closure of a set with respect to the topology , respectively, and they are defined by intersection of all -semiclosed and -closed sets containing , respectively. Co  represents the complements of members of . Moreover, a set of a bitopological space is -semi generalized closed (resp., -generalized semiclosed, -semi star generalized closed [21–23]) if - (resp., -, -) whenever and is -semiopen (resp., -open, -semiopen) in and .

-semi generalized closed sets, -generalized semiclosed sets, and -semi star generalized closed sets are denoted by - closed sets, - closed sets, and - closed sets, respectively.

3. Relations between Some Generalized Closed Sets

In this section, we discuss some relations between generalized closed sets in the bitopological spaces associated to the digraphs.

-open (resp., -open) sets and - closed sets are independent for and in general. For example, let . Then is -open but neither - closed nor - closed in . Also, is both - closed and - closed, but not -open in . Similarly, is -open but neither - closed nor - closed in . Also is both - closed and - closed, but not -open in .

Similarly, -closed (resp., -closed) sets and - closed sets are independent for = and in general. Since every in a bitopological space is associated to the digraph and every -open set is - open in every bitopological space , we have every -closed set is - open in for and . Also, every -closed set is - closed in and hence every -open set is - closed in associated to the digraph for and .

Suppose that is -open in . Then is -closed and hence it is -closed in . Also is -closed and hence is -open in . This implies that is -closed in associated to the digraph for and . So we have the following.

Theorem 3.1. Every -open (resp., -open) set is both - closed and - open in associated to the digraph for and .

Theorem 3.2. Every -closed (resp., -closed) set is both - closed and - open in associated to the digraph for and .

Since every - closed (resp., - open) sets are - closed, - closed and - closed (resp., - open, - open and - open) in , one can obtain the following:

Theorem 3.3. Every member of both and is - closed, - closed, - closed, - open, - open and - open, in associated to the digraph for and .

A subset of a bitopological space is -nowhere dense (resp., -somewhere dense) if -- (resp., --). Clearly, -nowhere dense sets and - closed sets are independent for and in general. For example, let . Then is - closed but not -nowhere dense in . Also, is - nowhere dense but not - closed in .

Suppose that is -nowhere dense in a bitopological space associated to the digraph . Then --. Since , one has -. This implies that . Hence, is - closed, - closed, - closed, - closed, - open, - open, - open, and - open in associated to the digraph for and .

Therefore, one can conclude that every nonempty - closed (resp., - closed, - closed, - closed, - open, - open, - open, and - open) set is -somewhere dense in associated to the digraph for and .

Since the set -cl is the least -open set containing in the bitopological space associated to the digraph , -cl whenever and is -open, for and . Hence every subset of the digraph is - closed and hence - open.

4. Conclusion

Thus, we have discussed the nature and properties of some generalized closed sets in the bitopological spaces associated to the digraphs in this short communication. This may be a new beginning for further research on the study of generalized closed sets in the bitopological spaces associated to the directed graphs. Hence, further research may be undertaken towards this direction. That is, one may take further research to find the suitable way of defining the bitopological spaces associated to the digraphs by using bitopological generalized closed sets such that there is a one-to-one correspondence between them. It may also lead to the new properties of separation axioms on these spaces.

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