Somewhat precontinuous mappings via grill
A. Swaminathany
Department of Mathematics
Government Arts College(Autonomous)
Kumbakonam, Tamil Nadu-612 002, India. and
M. Sankari
Department of Mathematics
Lekshmipuram College of Arts and Science Neyyoor,Kanyakumari
Tamil Nadu-629 802, India.
Abstract
This article introduces the concepts of somewhat G-precontinuous mapping and
somewhat G-preopen mappings. Using these notions, some examples and few interesting
properties of those mappings are discussed by means of grill topological spaces.
2010 Mathematics Subject Classi cation: 54A10, 54A20
Keywords: G-continuous mapping, somewhat G-continuous
mapping, G-precontinuous mapping, somewhat G-precontinuous mapping, somewhat
G -preopen mapping. G-predense set.
1 Introduction and Preliminaries The study of somewhat continuous functions was rst initiated by Karl.R.Gentry et al
in [4]. Although somewhat continuous functions are not at all continuous mappings it has y
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been studied and developed considerably by some authors using topological properties.
In 1947, Choquet [1] established the notion of a grill which has been milestone of
developing topology via grills. Almost all the foremost concepts of general topology
have been tried to a certain extent in grill notions by various intellectuals.
It is widely
known that in many aspects, grills are more e ective than a certain similar concepts
like nets and lters. E.Hatir and Jafari introduced the idea of G-continuous functions
in [3] and they showed that the concept of open and G-open are independent of each
other. Dhananjay Mandal and M.N.Mukherjee studied the notion of G-precontinuous
mappings in [2]. Our aim of this paper is to introduce and study new concepts namely
somewhat G-precontinuous mapping and somewhat G-semiopen mapping. Also, their
characterizations, interrelations and examples are studied.
Throughout this paper, Xstands for a topological space with no separation axioms
assumed unless explicitly given. For a subset Hof X, the closure of Hand the interior
of Hdenoted by Cl ( H) and Int ( H) repectively. The power set of Xdenoted by P(X ) .
The de nitions and results which are used in this paper concerning topological and grill
topological spaces have already taken some standard shape. We recall those de nitions
and basic properties as follows:
De nition 1.1. A mapping f: (X;F )! (Y ;F 0
) is called somewhat continuous[4] if there
exists an open set A, on (X ;F ) such that A f
1
(B ), for any open set B,
on (Y ;F 0
) .
De nition 1.2. A non-empty collection Gof subsets of a topological spaces X is said to