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Poopsilon
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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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Poopsilon said:I'm confused, so there are such open compact sets, but only in non-connected metric spaces?
deluks917 said:A compact subset of a metric space is closed.
Bacle2 said: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, 1/2^{n}), but then X cannot be complete, since m is not in X.
A compact set of metric spaces that is also open is a subset of a metric space that is both compact and open. This means that the set contains all of its limit points and is also contained within an open ball of finite radius centered at each point.
A compact set of metric spaces that is also open is a subset of a metric space that is both compact and open. This is different from just a compact set, which only requires the set to contain all of its limit points and does not necessarily have to be contained within an open ball of finite radius.
No, a compact set of metric spaces that is also open cannot be unbounded. By definition, a compact set must be bounded, and if it is also open, it cannot extend infinitely in any direction.
An example of a compact set of metric spaces that is also open is the closed interval [0,1] in the metric space of real numbers. This set contains all of its limit points and is also contained within an open ball of finite radius centered at each point.
A compact set of metric spaces that is also open has significant implications in mathematics, particularly in analysis and topology. It allows for the development of important theorems and concepts, such as the Heine-Borel theorem, which states that a subset of Euclidean space is compact if and only if it is both closed and bounded.