Examples of strongly rigid countable (semi)Hausdorff spaces

Abstract

A topological space $X$ is $strongly$ $rigid$ if each non-constant continuous map $f:X\to X$ is the identity map of $X$. A Hausdorff topological space $X$ is called $Brown$ if for any nonempty open sets $U,V\subseteq X$ the intersection $\bar U\cap\bar V$ is infinite. We prove that every second-countable Brown Hausdorff space $X$ admits a stronger topology $τ'$ such that $X'=(X,τ')$ is a strongly rigid anticompact Brown space.This construction yields an example of a countable anticompact Hausdorff space $X$ which is strongly rigid, which answers two problems posed at MathOverflow. By the same method we construct a strongly rigid $k_2$-metrizable semi-Hausdorff space containing a non-closed compact subset, which answers two other problem posed at MathOverflow.