A non
empty subset S of an [H.sub.v]-group (H, [omicron]) is called [H.sub.v]-subgroup of H if (S, [omicron]) is an [H.sub.v]-group.
For any non-
empty subset S of a valued field (Kv) with vK=Z we define:
Forward selection search [4] starts from an
empty subset [S.sub.x].
A non
empty subset S of a ring R is a completely prime ideal iff the characteristic function \s is a neutrosophic soft completely prime ideal over (R, E) where \s : E [right arrow] NS(R) is defined by :
Suppose ([c.sub.J](A) [intersection] [M.sub.J]) [intersection] (X - U) is a non
empty subset. Then ([c.sub.J](A) [intersection] [M.sub.J]) [intersection] (X - U) [subset] [c.sub.J](A) [intersection] (X - U) [subset] [c.sub.J](A) [intersection] (X-A) [subset] [c.sub.J](A) [intersection] A.
Example 15 Let X = {a,b,c,d}, [[phi].sub.N], [X.sub.N] be any types of the universal and
empty subset, and [A.sub.1] = <{a}, {b}, {c}> [A.sub.2] = <{a}, {b, d}, {c}>, then the family [GAMMA] = {[[phi].sub.N],[X.sub.N],[A.sub.1],[A.sub.2]} is a neutrosophic crisp topology on X.
Let (X, [tau]) be any vg-[C.sub.i] space and A be any non
empty subset of X then A is vg-[C.sub.i] iff ([A.sub.T/A]) is vg-[C.sub.i].
Theorem 4 Let [G.sup.c] be a c-edge-colored graph with no PEC closed trails, s, t [member of] V([G.sup.c]) and a non-
empty subset [PSI] of V ([G.sup.c]) \{s, t}.
Let <R [union] I> be any neutrosophic ring, a non
empty subset P of <R [union] I> is defined to be a neutrosophic ideal of <R [union] I> if the following conditions are satisfied;