Covering a Banach space

A well-known theorem by H. Corson states that if a Banach space admits a locally finite covering by bounded closed convex subsets, then it contains no infinite-dimensional reflexive subspace. We strengthen this result proving that if an infinite-dimensional Banach space admits a locally finite covering by bounded ω-closed subsets, then it is c₀-saturated, thus answering a question posed by V. Klee concerning locally finite coverings of l₁ spaces. Moreover, we provide information about massiveness of the set of singular points in (PC) spaces.