Almost Strongly Nncθe-continuous Functions

We introduce and investigate a new class of functions called almost strongly Nnceθ-continuous functions via Nnce-open sets in Nnc topological spaces. Also, some equivalent condition of almost strongly Nnceθ-continuous functions are proved.

The following are the quick consequence of Definition 2.2.
Definition 2.3 [15] A neutrosophic crisp topology (briefly, ncts) on a non-empty set P is a family τ of nc subsets of P satisfying the following axioms Then (P, τ ) is a neutrosophic crisp topological space (briefly, ncts ) in P . The τ elements are called neutrosophic crisp open sets (briefly, ncos) in P . A ncs C is closed set (briefly, nccs) iff its complement C c is ncos.
Definition 2.4 [5] Let P be a non-empty set. Then nc τ 1 , nc τ 2 , · · · , nc τ N are N -arbitrary crisp topologies defined on P and the collection N nc τ = {S ⊆ P : S = ( H j , L j ∈ nc τ j } is called N neutrosophic crisp (briefly, N nc )-topology on P if the axioms are satisfied: Then (P, N nc τ ) is called a N nc -topological space (briefly, N nc ts) on P . The N nc τ elements are called N nc -open sets (N nc os) on P and its complement is called N nc -closed sets (N nc cs) on P . The elements of P are known as N nc -sets (N nc s) on P .
Definition 2.5 [5] Let (P, N nc τ ) be N nc ts on P and H be an N nc s on P , then the N nc interior of H (briefly, N nc int(H)) and N nc closure of H (briefly, N nc cl(H)) are defined as The complement of an N nc ros (resp. N nc Sos, N nc Pos, N nc αos, N nc βos, N nc aos & N nc γos) is called an N nc -regular (resp. N nc -semi, N nc -pre, N nc -α, N nc -β, N nc aos & N nc -γ) closed set (briefly, N nc rcs (resp. N nc Scs, N nc Pcs, N nc αcs, N nc βcs, N nc acs & N nc γc)) in P .
The family of all N nc ros (resp. N nc rcs, N nc Pos, N nc Pcs, N nc Sos, N nc Scs, N nc αos, N nc αcs, N nc βos, N nc βcs, N nc aos, N nc acs, N nc γos & N nc γcs,) of P is denoted by N nc ROS(P ) (resp. N nc RCS(P ), N nc POS(P ), N nc PCS(P ), N nc SOS(P ), N nc SCS(P ), N nc αOS(P ), N nc αCS(P ), N nc βOS(P ), N nc βCS(P ), N nc aOS(P ), N nc aCS(P ), N nc γOS(P ) & N nc γCS(P )).  [20] A set H is said to be a Definition 2.7 [20] A set H is said to be a The complement of an N nc δos (resp. N nc δPos, The family of all N nc δPos (resp. N nc δPcs, N nc δSos, N nc δScs, N nc δαos, N nc δαcs, The N nc semiclosure, N nc preclosure, N nc b-closure and N nc α-closure are similarly defined and are denoted by N nc Scl(S), N nc Pcl(S), N nc bcl(S) and N nc αcl(S) respectively.

3.
Almost strongly N nc θ e-continuous functions Definition 3.1 Let S be a N nc set on a N nc ts P is said to be N nc e-regular (briefly, N nc er) if it is N nc eo and N nc ec.
The set of all N nc eθ-cluster points of S is called the N nc eθ-closure of S and is denoted by N nc ecl θ (S). A subset S is said to be N nc eθ-closed (briefly, N nc eθc) if S = N nc ecl θ (S). The complement of an N nc eθc set is called an N nc eθ-open (briefly, N nc eθo) set. Also it is noted in that

Definition 3.2
The family of N nc ro sets of a space (P, N nc τ ) forms a base for a smaller topology Lemma 3.1 Let P be a N nc ts. If S is a N nc Po set in P , then N nc scl(S) = N nc int(N nc cl(S)). Lemma 3.2 Let P be a N nc ts and S ⊆ P and {S α |α ∈ Λ} ⊆ P(P ). Then the following statements hold:  Proof. It can be proved directly using Lemma 3.2.

Definition 3.4
Let S be a subset of a N nc ts P . The N nc eθ-frontier (briefly, N nc eθF r ) of S is defined by N nc eF r θ (S) = N nc ecl θ (S)\N nc eint θ (S).

Theorem 3.2
The set of all points p ∈ P at which a function h : P → Q is not astN nc eθCts coincides with the union of the N nc eθF r of the inverse images of N nc ro sets of Q containing h(p).