Fuzziness in Chang's fuzzy topological spaces

Abstract

It is known that fuzziness within the concept of openness of a fuzzy set in a Chang's fuzzy topological space (fts) is absent. In this paper we introduce a gradation of openness for the open sets of a Chang jts (X, $\mathcal{T}$) by means of a map $\sigma\;:\; I^{x}\longrightarrow I\left(I=\left[0,1\right]\right)$, which is at the same time a fuzzy topology on X in Shostak 's sense. Then, we will be able to avoid the fuzzy point concept, and to introduce an adeguate theory for $\alpha$-neighbourhoods and $\alpha-T_{i}$ separation axioms which extend the usual ones in General Topology. In particular, our $\alpha$-Hausdorff fuzzy space agrees with $\alpha${*} -Rodabaugh Hausdorff fuzzy space when (X, $\mathcal{T}$) is interpreservative or $\alpha$-locally minimal.