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.
Copyright
Copyright © 1991 Hindawi Publishing Corporation. 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.
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