On regular κ-bounded spaces admitting only constant continuous mappings into T1 spaces of pseudocharacter ≤κ.

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 pseudocharacter ≤ κ , each continuous function f : R κ → Y is constant. This result resolves two problems posted by Tzannes [13] and extends results of Ciesielski and Wojciechowski [4] and Herrlich [8]. [ABSTRACT FROM AUTHOR]