A New Class on Ngα-Quotient Mappings in Nano Topological Space

The primary intend of this article is to define a new class of mappings called Ng αquotient mappings in nano topological space. The intention is to analyze characterizations and inter relationship of Ng α-quotient mappings with nano Tg#αspace, Ngα-continuous, Ngα-open, Ngα-irresolute, Ngα-homeomorphism and nano α-quotient mapping. Also several properties of strongly Ngα-quotient mapping are derived and the relationships among them are illustrated with the help of examples. Their interesting composition with strongly Ngα-irresolute are established. The concept of Ngα*-quotient mapping is explored and composition of mappings under strongly Ng α-quotient mapping and Ngα*-quotient mapping are discussed. Furthermore, to emphasize Ngα*-quotient mapping a few examples are considered and derived in detail.


Introduction
The class of quotient mappings is one of the most important classes of mappings in topology. Bhuvaneshwari et al. presented nano g-closed sets in nano topological space [1]. M. K. Gosh introduced separation axioms and graphs of functions in nano topological spaces [2]. Thivagar et al. established nano continuity in 2013 [3]. In another article the author introduced a nano topological space with respect to a subset X of an universe which is defined in terms of lower approximation, upper approximation and boundary region [4]. Thivagar et al. also defined quotient mappings in topological space [5]. The nano α-continuity was introduced by Nachiyar and K. Bhuvaneswari [6]. Nono et al. introduced the concept of g # -closed sets to investigate some of their topological properties [7]. The primary intention of this article is to introduce Ng # α -quotient mappings in nano topological spaces. Likewise, the concept of strongly Ng # α-quotient mapping and N # α*-quotient mapping are explored to study their fundamental properties in nano topological space.

Preliminaries
Definition 2.1 ( [8]). Let U be a non-empty finite set of objects called the universe and R be an equivalence relation on U named as the indiscernibility relation. Elements belonging to the same equivalence class are said to be indiscernible with one another. The pair (U, R) is said to be the approximation space and Let G ⊆U. Then (1). The lower approximation of G with respect to R is the set of all objects, which can be for certain classified as G with respect to R and it is denoted by (G). That is, where R(a) denotes the equivalence class determined by a. (2). The upper approximation of G with respect to R is the set of all objects, which can be possibly classified as G with respect to R and it is denoted by (G). That is, (G) = {R(a): R(a) G ≠ ∅, a U}. (3). The boundary region of G with respect to R is the set of all objects, which can be classified neither as G nor as not G with respect to R and it is denoted by (G). That is, (G) = (G) -(G).
Proof. Since p is N # α-homeomorphism, p is bijective and p is N # α-continuous. Let Definition 3.22. Let (U, τ R (G)) and (V, σ R ( )) be a nano topological spaces, then h: is nano open in (U, ( )) and hence the composite map h∘j is strongly N # α-irresolute.
Theorem 4.11. The composition of two N # α*-quotient mappings is also a N # *quotient mapping.

Conclusion
The current study yields Ng # -quotient mappings in nano topological space. The relationships among these introduced concepts are illustrated and their relationships with some nano topological notions such as N # α-irresolute, nano α-quotient mapping are shown with the help of examples. The notion of strongly N # α-irresolute function was presented which was helpful to verify the interesting results such as theorem 3.23. Some findings concerning strongly N # α-quotient mapping and enriched N # α*quotient mappings are investigated and illustrated with a few of their examples in detail. The presented concepts in this study are fundamental for further researches and will open a way to improve more applications on nano topology.