Compact Exhaustions and Open Tightness in Function Spaces

The compact-open function space maps compact subsets of a domain to basic neighborhoods in a function space. In this note, we investigate the way in which a compact exhaustion of a locally compact space influences the convergence properties of the compact-open function space . For a locally compact and σ-compact Hausdorff space , it is shown that each moving-off family of nonempty compact subsets of contains a countable strongly moving-off subfamily, from which it follows via a known characterization that the compact-open function space has open countable tightness. The proof is elementary; it separates the role of local compactness from that of a countable compact exhaustion. Examples include locally compact Lindelöf spaces and Euclidean domains. The result is stated as a straightforward corollary of known function-space theory, not as the announcement of a new characterization.