Some Properties of -Locally Closed Sets

A new kind of generalization of (1, 2)*-closed set, namely, (1, 2)*-locally closed set, is introduced and using (1, 2)*-locally closed sets we study the concept of (1, 2)*-LC-continuity in bitopological space. Also we study (1, 2)*-contracontinuity and lastly investigate its relationship with (1, 2)*-LC-continuity. 1. Introduction and Preliminaries It is established that generalization of closed set plays an important role in developing the various concepts in both topological and bitopological spaces. The difference of two closed subsets of an -dimensional Euclidean space was considered by Kuratowski and Sierpinski [1] in 1921 and the fundamental tool in their work is the notion of a locally closed subset of a topological space . In 1963 Kelly [2] initiated the systematic study of bitopological spaces. Then in 1989 Ganster and Reilly [3] used locally closed sets to define LC-continuity in a topological space. According to them a function is said to be LC-continuous if the inverse image of every open set in is locally closed set in . In 1990？ Jeli？ [4] introduced (1, 2)-locally closed sets and (1, 2) LC-continuity in bitopological space. In 1991？ Lellis Thivagar introduced the open set in bitopological space which is called (1, 2) open set [5]. In general we know that a (1, 2)*- -open set [6] may not be a -open set in , but in this present paper we have a necessary and sufficient condition for the requirement that an (1, 2)*- -open set is a -open set. Bourbaki [7] defined that a subset of a topological space is said to be locally closed if it is the intersection of an open and a closed subsets of in the year 1966. In 2004 M. Lellis Thivagar and O. Ravi introduced a generalized concept of (1, 2) open sets which is called (1, 2)*-open sets [8] in bitopological space. Using the (1, 2)*-open set and its complement in this paper we introduce (1, 2)*-locally closed set and (1, 2)*-separated set that is defined to obtain an improved result which gives that union of any two (1, 2)*-locally closed sets is again a (1, 2)*-locally closed set. We also established a relationship of (1, 2)*-regular open set [9] with (1, 2)*-Locally closed set. As an application of (1, 2)*-locally closed set we study (1, 2)*-LC-continuity. Lastly we introduce (1, 2)*-contracontinuous function and we investigate its relationship with (1, 2)*-LC-continuity. Throughout this paper , , and denote the bitopological spaces , , and , respectively, on which no separation axioms are assumed. The concept of (1, 2)*-continuous function from a bitopological space into another bitopological space