On Variations of Fuzzy Homogeneity

Abstract

Some of the important concepts in ordinary topological spaces like $n$ -homogeneity, strongly locally homogeneity, almost countable dense homogeneity, and transposition homogeneity are extended to fuzzy topological spaces and the relationship between these spaces together with fuzzy semi-discreteness is well studied here. Since there are spaces that fail to be fuzzy countable dense homogeneous ( $FCDH$ ) or fuzzy semi discrete ( $FSD$ ), only due to finitely many points, we found it necessary to study such spaces, which we refer to as almost fuzzy countable dense homogeneous ( $AFCDH$ ) and almost fuzzy semi discrete spaces ( $AFSD$ ) respectively. It is shown that the fuzzy extensions of the above notions are all fuzzy topological properties. A further variant of almost countable dense homogeneity called $AFCD{H}^{*}$ was introduced to fuzzy spaces and proved that a countable fuzzy separable space is $AFSD$ if and only if it is $AFCDH$ or $AFCD{H}^{*}$ . Hence, in countable fuzzy separable spaces, the notions $AFCDH$ and $AFCD{H}^{*}$ are equivalent. One of the main results identifies conditions under which an $AFCDH$ space becomes an $FCDH$ space. Further, the paper discusses ordinary topological spaces generated by fuzzy topological spaces and we shall observe that whenever the fuzzy topological space is almost fuzzy semi discrete, or fuzzy transposition homogeneous or almost fuzzy countable dense homogeneous, so is the corresponding ordinary space.