Examples of Strongly Rigid Countable (Semi)Hausdorff Spaces - Banakh, Taras; Stelmakh, Yarnya (Topology Proceedings, 2023)
A topological space $X$ is strongly rigid if each nonconstant continuous map $f: X → X$ 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 $\mathcal{T}^{′}$ such that $X^{′}=\left(X, \mathcal{T}^{′}\right)$ is a strongly rigid Brown space. This construction yields an example of a countable anticompact Hausdorff space $X$ which is strongly rigid. By the same method, we construct a strongly rigid semi-Hausdorff $k$-metrizable space containing a non-closed compact subset.