  We extend the theory of the Lusternik-Schnirelmann category to general metric
spaces by means of covers by arbitrary subsets. We also generalize the
definition of the strict category weight. We show that if the Bockstein
homomorphism on a metric space is non-zero, then its LS-category is at least
two, and use this to compute the category of Pontryagin surfaces. Additionally,
we prove that a Polish space with LS-category $n$ can be presented as the
inverse limit of ANR spaces of category at most $n$.
