Somewhat precontinuous mappings via grill Essay

Somewhat precontinuous mappings via grill

A. Swaminathany

Department of Mathematics

Government Arts College(Autonomous)

Kumbakonam, Tamil Nadu-612 002, India. and

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M. Sankari

Department of Mathematics

Lekshmipuram College of Arts and Science Neyyoor,Kanyakumari

Tamil Nadu-629 802, India.


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


(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

be a grill[1] on X if (i)

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