By the triplet (X, \tau, I), we mean an ideal topological space, where \tau is a topology on X, and I is an ideal on X i.e., I is a non-empty family of some subsets of X satisfying the conditions (i) \empty \in I, (ii) M \subseteq N and N \in I implies M \in I, and (iii) M \in I, N \in I implies M \cup N \in I.

Given M \subseteq X, M* is calculated as follows:

M* = \{x \in X : U_x \cap M \notin I for every open set U_x containing x\}, called local function of M.

M \subseteq X is called *-perfect if and only if M* = M.

These are prerequisite results that will help to give answer of the question:

Suppose A \subseteq (X, \tau, I) be such that A is non-empty and A = U \cap V, where U is open and V is *-perfect. Is it true that A \subseteq A* ?

For more results and deep concept related to M*, please see the reference:

D. Jankovic and T. R. Hamlett, New topologies from old via ideals, Amer. Math. Monthly, 97 (1990), 295-310.

Thanks in advance.