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This paper contains an equivalent statements of a pre-T2 space, where  = {(x, x) x  X} and K = {(x1, x2) XX  f(x1) = f(x2)} are considered subsets of XX with the product topology. An equivalence relation between the preclosed set  and a pre-T2 space, and a relation between a pre-T2 space and the preclosed set K with some conditions on a function f are found. In addition, we have proved that the graph C of R is preclosed in XX, if X/R is a pre-T2 space, where the equivalence relation R on X is open. On the other hand, we introduce the definition of a pre-stable (γpre-stable) set by depending on the concept of a pre-neighborhood, where we get that every stable set is pre-stable. Moreover, we obtain that a pre-stable (γpre-stable) set is positively invariant (invariant), and we add a condition on this set to prove the converse. Finally, a relationship between, (i) a pre-stable (γpre-stable) set and its component (ii) a preT2 space and a (positively critical point) critical point, are gotten.

Results on a Pre-T_2 Space and Pre-Stability