1. Introduction and Preliminaries.
The idea of "Gem-Set" is defined as: for a topological space )X,T) , and A?X, we defined A*x with
respect to space )X,T) as follows: A*x = {y ? X : G ? A ? Ix ,for every G ? T(y)} where T(y) = {G ? T : y ? G}, Ix
is an ideal on a topological space (X,T) at point x is defined by Ix = {U?X : x?Uc},where U is non-empty set of X.
Within this paper "Gem-Set" is studied with some its properties, a set of new separation axioms in
topological spaces, namely "I*-T0-space", "I*-T1-space", "I*-T2-space", "I**-T0-space", "I**-T1-space","I**-T2-space"
and the axioms are proposed by using the idea of "Gem-Set", the relationship between them is
studied. Also two mappings " I*- map " and" I**- map " are defined to carry properties of "Gem-Set" from a space to
other space.
Throughout this paper, spaces means topological spaces on which no separation axioms are assumed
unless otherwise mentioned. Let A be a subset of a space X. The closure and the interior of A are denoted by cl(A)
and int(A),