International Journal of Mathematics and Mathematical Sciences
Volume 13 (1990), Issue 3, Pages 513-516
doi:10.1155/S0161171290000746
A note on maximally resolvable spaces
Department of Mathematics, University of Patras, Patras, Greece
Received 23 November 1988; Revised 2 June 1989
Copyright © 1990 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
A.G. El'kin [1] poses the question as to whether any uncountable cardinal number can be the dispersion character of a Hausdorff maximally resolvable space.
In this note we prove that every cardinal number ℵ≥ℵ1 can be the dispersion character of a metric (hence, maximally resolvable) connected, locally connected space. We also proved that every cardinal number ℵ≥ℵ0 can be the dispersion character of a Hausdorff (resp. Urysohn, almost regular) maximally resolvable space X with the following properties: 1) Every continuous real-valued function of X is constant, 2) For every point x of X, every open neighborhood U of x, contains an open neighborhood V of x such that every continuous real-valued function of V is constant. Hence the space X is connected and locally connected and therefore there exists a countable connected locally connected Hausdorff (resp. Urysohn or almost regular) maximally resolvable space (not satisfying the first axiom of countability).