A note on maximally resolvable spaces (Q917964)

Summary: \textit{A. G. El'kin} [Vestn. Mosk. Univ. Ser. I 24, No.4, 66-70 (1969; Zbl 0183.512)] 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 \(\aleph \geq \aleph_ 1\) can be the dispersion character of a metric (hence, maximally resolvable) connected, locally connected space. We also prove that every cardinal number \(\aleph \geq \aleph_ 0\) 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).