SOME NEW CONCEPTS IN SOFT NANO TOPOLOGICAL SPACES

The notions of-soft nano subspaces, soft nano-closure and soft nano interior-in soft nano topological spaces are-introduced. Also, a new classes of sets namely weakly-soft nano g-closed sets, weakly soft nano-g-open sets and corresponding-closure and interior are introduced and their properties are investigated.. Further, the inter-relationship between-these new classes of soft nano sets with existing soft nano sets in soft nano topological spaces-are-studied.


INTRODUCTION
Soft set theory was introduced by Molodtsov [12] to overcome the drawbacks of theory ofprobability, the interval-mathematics and-theory of-fuzzy sets. A soft set over a universal set U is a structure (F, E) such that F: E→P(U), where E is a-parameter set and P(U) is the power set of U.
Shabir and Naz [13] introduced-the concept of soft topological-spaces and studied-the notions of soft-open sets, soft closed-sets, soft closure, soft-interior, soft-neighbourhood of a point and soft separation axioms. Theoretical studies and developments are made by many researchers in the 3171 SOME NEW CONCEPTS IN SOFT NANO TOPOLOGICAL SPACES field of soft sets and soft topological spaces [1], [3], [7], [8], [9], [10], [11]. Thivagar [14] introduced the concept-of nano-topology, using approximation spaces. Based on these backgrounds, Benchalli et al. [7] initiated the notion of soft nano-topological spaces, with the utilization of-soft set equivalence relation on the universal set and soft approximation spaces on a soft subset of the universal set.
Let U be the initial universal set, E is the set of parameters whose elements are attributes, characteristics or properties of the objects in U. Then, [4] the triplet (U, R, E), where R is a soft equivalence relation on U, is said to be the-soft approximation space for any subset X of U if: (i) The-soft lower approximation-of X corresponding to R and E is the set-of all objects,  In continuation, Benchalli et al. [5], [6]  In the present work, the notions of soft nano subspaces, soft nano-closure and soft-nano interior in soft-nano subspaces are initiated. The pivotal objective is to propose the concepts of weakly soft-nano g-closed sets, weakly soft-nano g-open sets and their properties are investigated.
Further, the inter-relationship between these new classes of soft nano-sets with existing soft nanosets in soft nano-topological spaces are studied. In addition, the definitions of weakly soft-nano gneighbourhood of a point, weakly-soft nano g-neighbourhood of a set, weakly soft nano-g-interior points and weakly soft nano g-limit points are introduced.
Throughout this paper, let U denotes the initial universal set and SNO(U, E) denotes the family of-all soft nano open sets in soft nano topological space (τR(X), U, E).

WEAKLY SOFT NANO GENERALIZED CLOSED SETS
In this section, we define weakly soft nano generalized closed (in short WSNg-closed) sets by using soft nano open sets and study some properties of WSNg-closed sets.     Since (G, E) is both soft nano open and soft nano closed, from [5] it follows that, SNInt(G, E) = (G, E) and SNCl(G, E) = (G, E). Therefore, SNInt(SNCl(G, E)) = (G, E) and SNCl(SNInt(G, E)) = (G, E).