Abstract
This paper attempts to forward both soft topology and fuzzy soft topology with a pioneering analysis of their mutual relationships. With each soft topology we associate a parameterized family of fuzzy soft topologies called its t-pushes. And each fuzzy soft topology defines a parameterized family of soft topologies called its t-throwbacks. Different soft topologies produce different t-pushes. But we prove by example that not all fuzzy soft topologies are characterized by their t-throwbacks. The import of these constructions is that some properties stated in one setting can be investigated in the other setting. Our conclusions should fuel future research on both fuzzy soft topology and soft topology.
Highlights
This paper lies at the crossroads of three disciplines, namely, topology, fuzzy set theory, and soft set theory.A consensus has taken hold that topology as a welldefined mathematical discipline originates in the early1900s, antecedents like Euler’s 1736 paper on theSeven Bridges of Königsberg can be traced back some centuries
According to [37], a fuzzy soft set on X is a pair (f, E), where E consists of all the attributes that are needed to characterize the elements of X, and f is a mapping f : E ! F ðXÞ: The set of all fuzzy soft sets on X with attributes E will be denoted by FSSE ðXÞ or
First we show that any fuzzy soft topology on X induces a collection of soft topologies on X indexed by
Summary
This paper lies at the crossroads of three disciplines, namely, topology, fuzzy set theory, and soft set theory. Other disciplines like mathematical economics adopted this technique, e.g., in the form of the Bergstrom-Walker theorem [2, 3] that states that any continuous acyclic relation defined on a compact topological space has a maximal element. This set of sufficient conditions is necessary [4]. The study of soft compactness owes especially to Aygünoğlu and Aygün [15] and Zorlutuna et al [14], and more recently, Al-shami et al [20] have produced seven generalized classes of soft semi-compact spaces Separability axioms are another remarkable element in the recent development of soft topology. At the end of the paper we summarize notation and conventions used along the article to facilitate the reading of our results
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