On invariant sets topology


1 Department of Mathematics, Semnan University, P.O.Box. 35195-363, Semnan, Iran.

2 Young Researchers Club, Semnan Branch, Islamic Azad University, Semnan, Iran.


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 $T_0, T_1$ or $T_2$.