Is there an (more or less) established name for the class of topological spaces in which every compact subset is metrizable? This is true for example in (LF)-spaces (inductive limits of Frechet-spaces) [see for example the article "Metrizability of precompact subsets in (LF)-spaces" by Cascales/ Orihuela].
Topological Space in which every compact subset is metrizable
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general-topology
functional-analysis
terminology