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

摘要

We construct two countable, Hausdorff, almost regular spacesI(S),I(T)having the following properties: (1) Every continuous map ofI(S)(resp,I(T)) into every Urysohn space is constant (hence, bothspaces are connected). (2) For every point ofI(S)(resp. ofI(T)) andfor every open neighbourhoodUof this point there exists an openneighbourhoodVof it such thatV⫅Uand every continuous map ofVintoevery Urysohn space is constant (hence both spaces are locallyconnected). (3) The spaceI(S)is first countable and the spaceI(T)nowhere first countable. A consequence of the above is the constructionof two countable, (connected) Hausdorff, almost regular spaces with adispersion point and similar properties. Unfortunately, none of thesespaces is Urysohn.