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

Abstract

In this paper for each cardinal $\kappa$ we construct an infinite $\kappa$-bounded (and hence countably compact) regular space $R_{\kappa }$ such that for any $T_1$ space $Y$ of pseudo-character $\le \kappa$, each continuous function $f:R_{\kappa }\to Y$ 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.