More Functions Associated with Neutrosophic gs α *- Closed Sets in Neutrosophic Topological Spaces

The concept of neutrosophic continuous function was very first introduced by A.A. Salama et al. The main aim of this paper is to introduce a new concept of Neutrosophic continuous function namely Strongly Neutrosophic gs α * - continuous functions, Perfectly Neutrosophic gs α * - continuous functions and Totally Neutrosophic gs α * continuous functions in Neutrosophic topological spaces. These concepts are derived from strongly generalized neutrosophic continuous function and perfectly generalized neutrosophic continuous function. Several interesting properties and characterizations are derived and compared with already existing neutrosophic functions.


Introduction
The concept of Neutrosophic set theory was introduced by F. Smarandache [1] and it comes from two concept, one is intuitionistic fuzzy sets introduced by K. Atanassov's [2] and the other is fuzzy sets introduced by L.A. Zadeh's [3]. It includes three components, truth, indeterminancy and false membership function. R. Dhavaseelan and S. Jafari [4] has discussed about the concept of strongly generalized neutrosophic continuous function. Further he also introduced the topic of perfectly generalized neutrosophic continuous function. The real life application of neutrosophic topology is applied in Information Systems, Applied Mathematics etc.
ii. Intersection of two neutrosophic set Ⱥ and Ƀ is defined as iii. Union of two neutrosophic set Ⱥ and Ƀ is defined as

Definition 2.3: [5]
A neutrosophic topology (N eu T) on a non-empty set  is a family τ N eu of neutrosophic sets in  satisfying the following axioms, In this case, the ordered pair , τ N eu ð Þor simply  is called a neutrosophic topological space (N eu TS). The elements of τ N eu is neutrosophic open set N eu À OS ð Þ and τ N eu c is neutrosophic closed set N eu À CS ð Þ . Definition 2.4: [6] A neutrosophic set Ⱥ in a N eu TS , τ N eu ð Þis called a neutrosophic generalized semi alpha star closed set N eu gsα 2. N eu gsα * À continuous [7] if the inverse image of each neutrosophic closed set in , σ N eu ð Þis a N eu gsα * À closed set in , τ N eu ð Þ.
4. strongly neutrosophic continuous [4] if the inverse image of each neutrosophic set in , σ N eu ð Þis both N eu À OS and N eu À CS in , τ N eu ð Þ.
5. perfectly neutrosophic continuous [4] if the inverse image of each N eu À CS in , σ N eu ð Þis both N eu À OS and N eu À CS in , τ N eu ð Þ.
Therefore, f is strongly N eu gsα * À continuous.

Perfectly neutrosophic gsα * -continuous function Definition 4.1:
A neutrosophic function f : , τ N eu ð Þ! , σ N eu ð Þis said to be perfectly N eu gsα * À continuous if the inverse image of every N eu gsα * À CS in , σ N eu ð Þis both N eu À OS and N eu À CS (ie, N eu À clopen set) in , τ N eu ð Þ. Theorem 4.2: Every perfectly N eu gsα * À continuous is strongly N eu gsα * À continuous, but not conversely. Proof: Þbe any neutrosophic function. Let Ⱥ be any . Therefore, f is strongly N eu gsα * À continuous. But f is not perfectly N eu gsα * À continuous, because f À1 Ⱦ ð Þ is not both N eu À OS and Þ. Therefore, f À1 Ⱦ ð Þ is not both N eu À OS and N eu À CS in , τ N eu ð Þ. Theorem 4.4: Every perfectly N eu gsα * À continuous is perfectly neutrosophic continuous, but not conversely.