The present study investigates some novel categorical properties of soft sets. By combining categorical theory with soft set theory, a categorical framework of soft set theory is established. It is proved that the category SFun of soft sets and soft functions has equalizers, finite products, pullbacks, and exponential properties. It is worth mentioning that we find that SFun is both a topologic...

Algorithmic randomness was originally defined for Cantor space with the fair-coin measure. Recent work has examined algorithmic randomness in new contexts, in particular closed subsets of 2ω ([2] and [8]). In this paper we use the probability theory of closed set-valued random variables (RACS) to extend the definition of Martin-Löf randomness to spaces of closed subsets of locally compact, Haus...

Shabir and Naz in [25] introduced the notion of soft topological spaces. They defined basic notions of soft topological spaces such as open soft sets, closed soft sets, soft subspaces, soft closure, soft nbd of a soft point, soft separation axioms, soft regular spaces, soft normal spaces and they established their several properties. Min in [19] investigated some properties of such soft separat...

We deal with the new class of pre-ℐ-regular pre-ℐ-open sets in which the notion of pre-ℐ-open set is involved. We characterize these sets and study some of their fundamental properties. We also present other notions called extremally pre-ℐ-disconnectedness, locally pre-ℐ-indiscreetness, and pre-ℐ-regular sets by utilizing the notion of pre-ℐ-open and pre-ℐ-closed sets by which we obtain some eq...

In this article we construct closed, isospectral, non-isometric locally symmetric manifolds. We have three main results. First, we construct arbitrarily large sets of closed, isospectral, non-isometric manifolds. Second, we show the growth of size these sets of isospectral manifolds as a function of volume is super-polynomial. Finally, we construct pairs of infinite towers of finite covers of a...

Soft set theory, proposed by Molodtsov has been regarded as an effective mathematical tool to deal with uncertainties. Vague set is a set of objects, each of which has a grade of membership whose value is a continuous subinterval of [0, 1]. Such a set is characterized by a truth-membership function and a falsemembership function. Thus, a vague set is actually a form of fuzzy set, albeit a more ...