Intersection properties for coverings of G-spaces

The Ljusternik–Schnirelmann–Borsuk theorem for antipodal maps on the sphere can be stated as an intersection property of coverings of the sphere. We generalized this theorem to free finite-group actions on paracompact Hausdorff spaces and discuss how this result might be improved. We illustrate our results by a free action of the Klein four-group on the torus.