Hausdorff compactifications of topological function spaces via the theory of continuous lattices

It is known from the theory of continuous lattices that if X is a locally compact Hausdorff space then the set LSC(X) of lowersemicontinuous functions defined on X with values on the extended real lineadmits a unique compact Hausdorff topology making the functional (f, g)to min(f, g) continuous, namely the Lawson topology of the continuouslattice LSC(X). It is natural to wonder whether the relative topologyon the subset C(X) of continuous functions is the compact-opentopology. Unfortunately, it turns out to be strictly weaker. But a relatedconstruction does produce a Hausdorff compactification of C(X). Weshow that if X is a locally compact Hausdorff space and Y is aHausdorff topological space which is perfectly embedded into a continuouslattice L endowed with the Scott topology, then the Lawson topologyon the continuous lattice LSC(X,L) of Scott continuous maps fromX to L induces the compact-open topology on the spaceC(X,Y) of continuous maps from X to Y. Thus, by takingthe closure of the image of C(X,Y) in LSC(X,L), one gets aHausdorff compactification of C(X,Y). Three particular cases are ofinterest. (1) If Y is the Euclidean real line one can take L as the lattice of compact connected subsets of the two-pointcompactification of Y ordered by reverse inclusion. In this case,C(X,Y) is already dense in LSC(X,L). (2) If Y is alocally compact Hausdorff space, one can take L as the compactsubsets of the one-point compactification of Y. (3) As a furtherparticular case of (2), if X and Y are compact Hausdorff, oneconcludes that the Vietoris topology on the closed subsets of the cartesianproduct of X and Y induces the compact-open topology onC(X,Y), by identifying continuous functions with their closed graphs, using the fact that the Lawson topology coincides with the Vietoristopology.