IJPAM: Volume 44, No. 4 (2008)

SOME CLASSES OF GENERALIZED
COMPLETELY REGULAR SPACES

Bhamini M.P. Nayar
Department of Mathematics
Morgan State University
1700, E. Cold Spring Lane, Baltimore, MD 21251, USA
e-mail: Bhamini.Nayar@morgan.edu


Abstract.Studying spaces defined by functions into real number system, we study four classes of spaces, namely the classes of $SR$-spaces, $SER$-spaces, $SSR$-spaces and $\alpha -CR$-spaces. A space is said to be an $SR$-space (respectively, $SER$-space, $SSR$-space. $\alpha-CR-space$), if for each closed set $F \subseteq X$ and a point $x \in X - F$ there is a semi-continuous (respectively, upper semi-continuous and quasi lower semi-continuous, lower semi-continuous and quasi upper semi-continuous, upper semi-continuous and lower $\alpha$-continuous) function $f: X\rightarrow [0,1]$ such that $f(x) = 0$ and $f(F) = 1$. Each of these classes of spaces is characterized by characteristic functions obtained from different generalizations of continuous functions and has properties which are analogous to the properties of completely regular spaces. Hence each of these types of spaces is considered as a Generalized Completely Regular Space.

Received: March 10, 2008

AMS Subject Classification: 54D25, 54D55

Key Words and Phrases: completely regular, regular, semi-regular, strongly s-regular, $\alpha$-set, upper semi-continuous functions, lower semi-continuous functions

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2008
Volume: 44
Issue: 4