Suppose that V is a locally compact topological space that admits no countable covering with compact sets. Let <spanclass=′latex−bold′>C</span>
denote the set of all compact subsets of the space V and <spanclass=′latex−bold′>U</span> the set of open subsets that are not contained in any compact set. Let f be a function from <spanclass=′latex−bold′>U</span> to <spanclass=′latex−bold′>C</span> such that f(U)⊆U for all U∈<spanclass=′latex−bold′>U</span>. Prove that either
(i) there exists a nonempty compact set C such that f(U) is not a proper subset of C whenever C⊆U∈<spanclass=′latex−bold′>U</span>,
(ii) or for some compact set C, the set f−1(C)=⋃{U∈<spanclass=′latex−bold′>U</span>:f(U)⊆C} is an element of <spanclass=′latex−bold′>U</span>, that is, f−1(C) is not contained in any compact set.
A. Mate