Compactness in the fine topology (Q792650)

For Y metric and X Hausdorff, locally compact Lindelöf, the author characterizes compact subsets of the space of continuous maps from X to Y, topologized with the fine (Whitney) \(C^ 0\)-topology. For smooth manifolds similar characterization is given for compact subsets of the space of \(C^ r\) maps in the Whitney \(C^ 1\) topology. Roughly, \(Q\subset {\mathcal C}(X,Y)\) is compact if and only if Q is compact in the compact-open topology and is covered by a finite number of sets having a special form.