International Journal of Mathematics and Mathematical Sciences
Volume 14 (1991), Issue 4, Pages 709-714
doi:10.1155/S0161171291000959
Two countable Hausdorff almost regular spaces every contiunous map of which into every Urysohn space is constant
Department of Mathematics, University of Patras, Patras 26110, Greece
Received 13 July 1990; Revised 2 November 1990
Copyright © 1991 V. Tzannes. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.