[10] Let X be a

non empty set, then A = {(x, [[mu].sub.A](x), [[gamma].sub.A](x)) : x [member of] X} is called an intuitionistic set on X, where 0 [less than or equal to] [[mu].sub.A](x) + [[gamma].sub.A](x) [less than or equal to] 1 for all x [member of] X, [[mu].sub.A](x),[[gamma].sub.A](x) [member of] [0, 1] are the degree of membership and non membership functions of each x [member of] X to the set A respectively.

Finally, we need the following special function [pi] defined on the set, [P.sup.+], of

non empty partitions and the "Mobius-like" arithmetical function, [upsilon], defined on the set, [N.sup.+], of positive integers.

In a vg-regular space X, if X is a vg-[T.sub.0]-limit point of X, then for any

non empty vg-open set U, there exists a

non empty vg-open set V such that vg[bar.(V)] [subset] U, x [not member of] vg[bar.(V)].

Let X be a

non empty set and A [member of][[zeta].sup.X] Then [A.sup.~] = <[[mu].sub.A](x) , [[gamma].sub.A](x) > for every x [member of] X.

* [[beta].sub.i] and [[alpha].sub.i+1] must have the same height, for every i [not equal to] 1; if i = 1, we have that [[beta].sub.1] is always

non empty and the height of [[beta].sub.1] is equal to the width of [[alpha].sub.2] plus one;

[4] A collection G of

non empty subsets of a space X is called a grill on X

When the proportion of

non empty urns reaches r, one ball is removed in every

non empty urn.

An interval neutrosophic set [50] [??] of a

non empty set H is expreesed by truth-membership function [t.sub.[??]](h) the indeterminacy membership function [i.sub.[??]](h) and falsity membership function [f.sub.[??]](h).

Definition 2.4.[5] An intuitionistic fuzzy topology (IFT) in Coker's sense on a

non empty set X is a family [tau] of IFSs in X satisfying the following axioms.

Definition 2.1 [5] Let X be a

non empty set, then [tau].sub.1], [[tau].sub.2], ..., [[tau].sub.2] be N-arbitrary topologies defined on X and the collection [N.sub.[tau]] = {S [subset or equal to] X : S = ([[universal].sup.N.sub.i=1] [A.sub.i]) [union] ([[??].sup.N.sub.i=1] [B.sub.i]), [A.sub.i], [B.sub.i] [member of] [[tau].sub.i]} is, called a N-topology on X if the following axioms are satisfied:

A

non empty set X together with two topologies [[tau].sub.1] and [[tau].sub.2] is called a Bitopological space[2].

A neutrosophic crisp topology (NCT) on a

non empty set X is a family of [GAMMA] of neutrosophic crisp subsets in X satisfying the following axioms: