On regular $κ$-bounded spaces admitting only constant continuous mappings into $T_1$ spaces of pseudo-character $\leq κ$

In this paper for each cardinal $κ$ we construct an infinite $κ$-bounded (and hence countably compact) regular space $R_κ$ such that for any $T_1$ space $Y$ of pseudo-character $\leqκ$, each continuous function $f:R_κ\rightarrow Y$ is constant. This result resolves two problems posted by Tzannes in Open Problems from Topology Proceedings and extends results of Ciesielski and Wojciechowski and Herrlich.

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