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In this paper for each cardinal $\kappa$ we construct an infinite$\kappa$-bounded (and hence countably compact) regular space $R_{\kappa}$ suchthat for any $T_1$ space $Y$ of pseudo-character $\leq\kappa$, each continuousfunction $f:R_{\kappa}\rightarrow Y$ is constant. This result resolves twoproblems posted by Tzannes in Open Problems from Topology Proceedings andextends results of Ciesielski and Wojciechowski and Herrlich.

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