Two countable Hausdorff almost regular spaces every contiunous map of which into every Urysohn space is constant

Abstract

We construct two countable, Hausdorff, almost regular spaces I(S), I(T) having the following properties: (1) Every continuous map of I(S) (resp, I(T)) into every Urysohn space is constant (hence, both spaces are connected). (2) For every point of I(S) (resp. of I(T)) and for every open neighbourhood U of this point there exists an open neighbourhood V of it such that V⫅U and every continuous map of V into every Urysohn space is constant (hence both spaces are locally connected). (3) The space I(S) is first countable and the space I(T) nowhere first countable. A consequence of the above is the construction of two countable, (connected) Hausdorff, almost regular spaces with a dispersion point and similar properties. Unfortunately, none of these spaces is Urysohn.