Lusternik, Schnirelman for subspaces

We investigate for involutions σ :X→X colorings of the subspaces A, i.e., finite coverings A1,…,Ak with Ai∩σ(Ai)=∅. One of the results is that there is a difference of behaviour between the relative closed colorings, the relative open colorings and the open colorings of A. Upperbounds in terms of the dimension of the subspace are obtained and an example is provided that shows that the obtained upperbounds are best possible. Moreover, we construct an example with finite relative closed coloring number, finite relative open coloring number but infinite open coloring number and an example with finite relative closed coloring number but infinite relative open coloring number.