GLC AND GLC** CONTINUOUS FUNCTIONS: A CONCEPTUAL FLAW

The concept of generalized locally closed sets (glc-sets), GLC**-sets followed by the notion of GLC and GLC**continuous maps was initiated by Balachandran et al. (Generalized locally closed sets and GLC-continuous functions, Indian J. pure appl. Math 27(3): 235-244, 1996). In the present work, it has been established that the collection of glc-sets and the collection of GLC** -sets, each is exactly equal to the power set P (X) of X. Consequently, any arbitrary function with any choice of domain and range turns out to be GLC and GLC**-continuous function which is not desirable from analytic point of view.


INTRODUCTION
The idea of locally closed set was introduced by Bourbaki [2] in 1966. (see also [3]). This concept of locally closed set had been used by Ganster and Reilly [4] for defining the generalized version of continuity viz. LC-irresolute, LCcontinuity and sub-LC-continuity. Balachandran et al. [1] had extended the definition of locally closed sets and initiated the notion of "Generalized locally closed set", in particular, glcset, GLC*-set and GLC**-set. Since last few decades many topologist (cf. [4], [5], [6], [7], [8], [9], [10], [11]) are trying to explore the possibility of generalizing the classical phenomenon "continuity" of the function defined in the topological space. Following this trend Balachandran et al. [1] have also defined and explored the idea of GLC-irresolute maps and GLC-continuous maps. Extending the idea of Balachandran et al. [1], Park et al. [8] have defined semi generalized locally closed sets and locally-generalized closed sets along with SGLC-continuous functions and L GLCcontinuous functions respectively. (see also [9], [10], [11]). Recently, Patil et al. [10] have further extended the concept of glc-sets and introduced the notion of g*w -lc sets and g*w -lc sets and g*w -lc sets and have applied these concepts to define relevant different types of continuous functions. In the present paper, authors have established that the respective collection of glc-sets, and the collection of GLC**sets (cf. [1]) generated by the topology yield precisely the power set P(X) of X. This information leads to the conclusion that the corresponding GLC and GLC** -idea of continuity is not enhancing the class of continuous functions with some relaxed conditions however all functions with arbitrary domain and range turns out to be GLC and GLC**-continuous functions which is inadequate. In view of this observation, all the extensions turned out to be superfluous.

PRE-REQUISITES
The following notations have been referred throughout this work: (X, ) -Topological space with topology defined on the set X ( ). (A) -Closure of A for the subset A of X with respect to (X, ). (A)-Interior of A for the subset A of X with respect to (X, ). P(X) -Power set of X.
Definition 2.1. A subset B of (X, ) is called g-closed [12] if whenever for an open set in a topological space (X, ). A subset of (X, ) is called g-open if its complement is g-closed.  (V) GLC** ) for each V (cf. [1]).

MAIN RESULT
We are now set to state the main result of this paper. Theorem 3.1. Let (X, ) be the topological space and GLC and GLC** be the collection of sets described in the Definition 2.2 and 2.3 respectively. Then GLC GLC** P (X) where P (X) is the power set of X.
Proof. Let X be any non empty set = { , X, } be the topology on X. Let A be any non empty proper subset of X. The following cases have been considered: Therefore is g-closed.
ii) We next consider ( ) ∈ J such that for and = Thus, is g-closed.

Claim:
where is g-closed and is g-open (cf. Remark 2.1). Consider = = = Hence, (3.1) holds and finally we conclude that A is glc-set. Since, A was arbitrary subset of X, every subset of X is glc-set. Thus, the collection GLC is precisely equal to P (X).
Since (open), it is direct by the Definition 2.3 that GLC** P(X) This completes the proof.

CONCLUSION
• In view of Definitions 2.4 and 2.5, each function defined from (X, ) to (Y, ) turns out to be GLCcontinuous (irresolute) and GLC** -continuous (irresolute) which is not acceptable as a generalization of the classical concept of continuity in Topology.
• All generalizations of GLC-set and GLC** -set turned out to be stagnant and finally not desirable.