Are there (non-trivial) examples of topological spaces in which the closure of any path connected set is path connected?
If so, are there any far reaching topological consequences of this property? Can the spaces be described?
2026-04-18 04:22:13.1776486133
Examples of spaces in which the closure of any path connected set is path connected.
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Any totally disconnected space is an example (for example, the Cantor set). Indeed, the only path-connected subsets of it are singletons, which are already closed!
(Perhaps not so interesting, but hey, the Cantor set isn't exactly trivial! :))
I don't think that this property is particularity "characteristic". Path components can be pretty silly in an arbitrary space, because the topology can be quite incompatible with the topology of the interval. You might want to add some restrictions on the kind of space that you're looking for.