Results on a Pre-T_2 Space and Pre-Stability

Abstract

This paper contains an equivalent statements of a pre-  space, where  are considered subsets of with the product topology. An equivalence relation between the preclosed set  and a pre-  space, and a relation between a pre-  space and the preclosed set  with some conditions on a function  are found. In addition, we have proved that the graph  of  is preclosed in if  is a pre-  space, where the equivalence relation  on  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- space and a (positively critical point) critical point, are gotten.