Results on a Pre-T2 space and pre-stability

الملخص EN

This paper contains an equivalent statements of a pre-T2 space, where ={(x,x and ={(x1,x2)(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 with some conditions on a function 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 (pre-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 (pre-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 (pre-stable) set and its component (ii) a pre-T2 space and a (positively critical point) critical point, are gotten.