A note on maximally resolvable spaces. (1990)

• Record Type: Journal Article
• Authors: Tzannes, V.
• Is Part Of: International journal of mathematics and mathematical sciences. Volume 13:Number 3(1990)
• Page Start: 513
• Page End: 516
• Publication Date: 1990
• Subjects: metric -- countable spaces -- connected -- locally connected -- maximally resolvable -- Urysohn -- almost regular spaces; Mathematics -- Periodicals; 510.5
• DOI: 10.1155/S0161171290000746
• Languages: English
• ISSNs: 0161-1712

Abstract

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).