Are there conditions on a topological space $X$ under which its Lusternik-Shnirelman category is countable (or even finite)?
"Countable Lusternik-Shnirelman category" means that $X$ can be covered by countable many sets, each of them is contractible in $X$. (And "finite L-S category" means of course it's covered by finitely many such sets.)
I'm interested in topological spaces which are not necessarily CW-complexes.