On invariant sets topology

In this paper, we introduce and study a new topology related to a self mapping on a nonempty set. Let X be a nonempty set and let f be a self mapping on X. Then the set of all invariant subsets of X related to f, i.e. \tau_f := \{A\subseteq X : f(A)\subseteq  A\}\subseteq \mathcal{P}(X) is a topology on X. Among other things, we find the smallest open sets contains a point x\in X. Moreover, we find the relations between f and \tau_f . For instance, we find the conditions on f to show that whenever \tau_f is T_۰, T_۱ or T_۲.

کلیدواژه ها:

Topological spaces ، Separation axioms ، fixed point theorems