Resumen de Partitions of spaces by locally compact subspaces

Vitalij Chatyrko, Yasumao Hattori

• 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)