From Wikipedia
A metric space (or topological vector space) is said to have the
Heine–Borel property if every closed and bounded subset is compact.
Any subset of a Euclidean space, including itself, has the Heine–Borel property. I was wondering if there are more general types of metric spaces, topological vector spaces, or whatever space where boundedness and closedness can make sense, such that they also have the Heine–Borel property?
Or does the Heine–Borel property characterize subsets of Euclidean spaces?
Thanks and regards!
Best Answer
0) Note that in general a metric space has this property iff it is ball compact, i.e., closed balls of finite radius are compact.
Ball compact spaces are locally compact, but the converse does not hold: e.g. an infinite set endowed with the discrete metric $d(x,y) = \delta_{x,y}$ is locally compact, bounded and not compact, hence not ball compact.
1) A topological field is ball compact if and only if it is locally compact and not discrete. Thus the topological fields with this property are $\mathbb{R}$ and $\mathbb{C}$, $\mathbb{Q}_p$ and its finite extensions and $\mathbb{F}_q((t))$.
2) A finite product of ball compact metric spaces, endowed with (say) the product metric $d = \max_{i=1}^n d_i$ is ball compact.
Combining these, we find that any finite dimensional vector space over a nondiscrete locally compact field has this property. This directly generalizes the spaces $\mathbb{R}^n$ and there are branches of mathematics (number theory, representation theory, harmonic analysis) in which this generalization is very natural.