Partitions of spaces by locally compact subspaces

• Autores: Vitalij Chatyrko, Yasumao Hattori
• Localización: Houston journal of mathematics, ISSN 0362-1588, Vol. 32, Nº 4, 2006, págs. 1077-1091
• Idioma: inglés
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• Resumen

  • In this article, we shall discuss the possibility of different presentations of topological spaces as unions or partitions of locally compact subspaces.

    For a space X, let lc(X) (resp. lcd(X)) denote the minimum cardinality of a cover (resp. partition) of X by locally compact subspaces. We prove:

    (1) For every finite or countably infinite cardinal n, there exists a subspace X of the real line such that lc(X)=n; (2) for every Hausdorff space X, if lc(X) is at most countable, then lc(X)=lcd(X); (3) if X is a topologically complete, nowhere locally compact, Hasudorff space, then lc(X) is uncountable; (4) if a perfectly normal space X is covered by finitely many subpaces, each of which is locally compact with at most one exception, then the covering dimension of X coincides with the maximum of the dimensions of those subspaces. It is open whether there is an example in ZFC of a Hausdorff space X such that lc(X)