On β-Normal Spaces

Abstract

This thesis revisits some types of normal spaces, namely, β-normal spaces and α-normal spaces, which were introduced by Arkhangel'skii and Ludwig in 2001. We study some properties of these spaces with a focus on improving some of the already existing properties and exploring new properties that are not available in the literature. Under β-normal spaces, among other things, we characterize these spaces using some types of open sets. We use the ultrafilter space to answer Murtinova's question about the existence of a β-normal and regular space which is not Tychonoff. α-normal spaces are described in terms of countable open sets, a result imitating that of normality. It turns out that continuous functions which are onto, open and closed preserve β-normality, while those which are injective, open and closed reflect α-normality. The notion of β-normal spaces is extended to the category of bitopological spaces where we characterize these bitopological spaces simultaneously in terms of i-open sets, (i, j)-preopen and (i, j)-α-open sets. We study the interrelations of these spaces with other bitopological spaces.