Examples of Strongly Rigid Countable (Semi)Hausdorff Spaces
Keywords:
anticompact spaces, Brown space, $k$-metrizable space, rigid space, Rudin-Keisler incomparable ultrafilters, strongly rigid spaceAbstract
A topological space $X$ is strongly rigid if each nonconstant continuous map $f: X \rightarrow X$ 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 $\mathcal{T}^{\prime}$ such that $X^{\prime}=\left(X, \mathcal{T}^{\prime}\right)$ 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.
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