Could we define the terms locally connected/compact/contractible/simply-connected/whatever to mean that there is a basis (for the topology on our space) of connected/compact/contractible/simply-connected/whatever subsets? I mean does this usage of these terms agree with the "standard" ones?
2026-04-30 08:36:51.1777538211
Local topological properties
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No, this doesn't always work. For instance, $\mathbb R$ is a locally compact space, but it has no basis composed of compact sets simply because the only compact and open set in $\mathbb R$ is the empty set.