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Are there any down to earth examples besides the empty set?
Edit: No discrete metric shenanigans either.
Edit: No discrete metric shenanigans either.
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I'm confused, so there are such open compact sets, but only in non-connected metric spaces?
A compact subset of a metric space is closed.
Deluks917 wrote, in part:
"A compact subset of a metric space is closed"
Just a quick comment, to refresh my metric topology: an argument for why compact subsets X of metric spaces M are closed: as metric subspaces, compact metric spaces are complete. Now, use closed set version that a subset is closed iff (def.) it contains all its limit points. Now, assume there is a limit point m of X that is not in X. Then you can construct a Cauchy sequence in X that would converge to m, e.g., by taking balls B(m, 1/2n), but then X cannot be complete, since m is not in X.