Minimal covers of open manifolds with half-spaces and the proper L-S category of product spaces (Q1420720)

With an adaptation of the notion of contractibility at the proper homotopy setting, the authors define the proper Lusternik-Schnirelmann category of a locally compact, \(\sigma\)-compact Hausdorff space. In [Manuscr. Math. 29, 385--415 (1979; Zbl 0415.55001)], \textit{W. Singhof} gives a sufficient condition for having an equality between the classical LS-category of a manifold \(M\) and the smallest number of balls needed to cover \(M\). Here the authors obtain a similar result between the proper LS-category and the smallest number of half-spaces needed to cover \(M\). They also study the behaviour of their invariant with respect to the product of spaces.