A note on maximally resolvable spaces. (1990) - Tzannes, V.
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 numberℵ ≥ ℵ 1 can be the dispersion character of a metric (hence, maximally resolvable) connected, locally connected space. We also proved that every cardinal numberℵ ≥ ℵ 0 can be the dispersion character of a Hausdorff (resp. Urysohn, almost regular) maximally resolvable spaceX with the following properties: 1) Every continuous real-valued function ofX is constant, 2) For every pointx ofX, every open neighborhoodU ofx, contains an open neighborhoodV ofx such that every continuous real-valued function ofV is constant. Hence the spaceX 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).