A note on maximally resolvable spaces

V. Tzannes

International Journal of Mathematics and Mathematical Sciences, 1990, vol. 13, 1-4

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 ≥ ℵ 1 can be the dispersion character of a metric (hence, maximally resolvable) connected, locally connected space. We also proved that every cardinal number ℵ ≥ ℵ 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).

Date: 1990
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:796870

DOI: 10.1155/S0161171290000746

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