Images of collections of proper maps which determine a topology

Let F be a collection of proper maps into a topological space, (X, τ), which satisfies a topological property P. F is said to fix the topology τ on X among all topologies satisfying property P if and only if τ is the only topology on X which satisfies property P and for which every element of F is continuous and proper. Let X be a set, F a collection of functions into X, and im(F) = ∪ƒ∈F{ƒ(J) | ƒ:J→X}. It will be shown that the following is true: “If F is a collection of proper maps which fixes the topology of a locally compact Hausdorff space (X, τ) among all locally compact Hausdorff topologies on X, then X − im(F) is a finite set of isolated points”. Thus in this context if X has no isolated points then im(F) = X. When P is the property: “first countable and Hausdorff”, the following criterion for a collection of proper maps to fix a 1st countable T2 space can be proven: “Let (X, τ) be a first countable Hausdorff space. Suppose F is a collection of proper maps from first countable Hausdorff spaces to X such that for each sequence in X, some element of F passes through a subsequence. Then F fixes τ among all first countable Hausdorff topologies on X”.