![]() ![]() Whether these are the only spaces that do, I don't know. So all metric spaces with the Heine-Borel property satisfy the requirements of your 3rd question. Regarding your third question, since in a metric spaces with the Heine-Borel property all bounded sequences have convergent subsequences, all bounded sequences also have Cauchy subsequences (because any convergent subsequence is trivially Cauchy). It is also true that any noncompact space $(X,d)$ has at least one unbounded metric $d'$ that preserves its topology, but this doesn't guarantee that if $(X,d)$ has the Heine-Borel property, then $(X,d')$ does too (because different metrics create different bounded sets). The metric spaces that satisfy the requirements of your second question are then exactly like euclidean space, in that they'll be noncompact spaces with the Heine-Borel property. As $A$ was an arbitrary closed bounded subset of $X$, all closed bounded subsets of $X$ are compact.īy the Heine-Borel theorem, all metric spaces $(\mathbb R^n,d)$ possess this Heine-Borel property (where $d$ can be either the standard Euclidean metric or the Chebyshev distance function). This makes $A$ sequentially compact, and hence compact. every sequence in $A$ has a convergent subsequence in $A$. The converse is also true, by which I mean if a metric space is sequentially compact, then it is also compact.įirst, suppose $(X,d)$ is a metric space in which every closed bounded subspace is compact. ![]() ![]() You know that if a metric space, $(X,d)$, is compact, then every sequence in $X$ has a convergent subsequence (this is known as sequential compactness). The category of metric spaces you are looking for are those in which every closed bounded subset is compact (this property is called the Heine-Borel property). ![]()
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