With each topological property P one can associate a relative version of it
formulated in terms of the location of Y in X in such a natural way that
when Y coincides with X, then this relative property coincides with P.
Arhangel?skii and Genedi introduced this concept of relative topological
properties in 1989. The concept of mild normality or ?-normality was
introduced independently by Singal and Singal in 1973 and Scepin in 1972.
A few years earlier in 1969, Singal and Arya studied the concept of almost
normality. V. Za?cev in 1968 introduced the concept of quasi normal spaces
while ?-normality was studied by Kalantan in 2008. In this paper we study
these variants of normality in a relative sense.
Every topological property can be associated with its relative version in such a way that when smaller space coincides with larger space, then this relative property coincides with the absolute one. This notion of relative topological properties was introduced by Arhangel’skii and Ganedi in 1989. Singal and Arya introduced the concepts of almost regular spaces in 1969 and almost completely regular spaces in 1970. In this paper, we have studied various relative versions of almost regularity, complete regularity, and almost complete regularity. We investigated some of their properties and established relationships of these spaces with each other and with the existing relative properties.
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