Lusternik-Schnirelman for subspaces (Q5946449)

The origin of this work is the following well-known theorem of Lyusternik and Shnirel'man: ``If the sphere \(S^n\) is covered by \(n+1\) open (or closed) sets, then one of these sets must contain an antipodal pair''. The authors address the following generalization: if \(X\) is a space equipped with an involution \(\sigma:X\to X\) and if \(A\) is a subspace of \(X\), what is the minimum size of a cover of \(A\) by sets \(A_i\) with \(A_i\cap \sigma(A_i)= \emptyset\)? One of the results is that there is a difference of behaviour depending if the sets \(A_i\) are required to be open in \(X\), open in \(A\) or closed in \(A\). Upper bounds are given and several examples are described.