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 X the intersection U V is infinite. We prove that every second-countable Brown Hausdorff space X admits a stronger topology T^′ such that X^′=(X, T^′) 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.