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I was looking at the Heine-Borel theorem. It boils down to the following two claims about a metric space $X$:

  • Every closed subspace of $X$ is complete. (1)
  • Every bounded subspace of $X$ is totally bounded. (2)

When both hold for $X$, every closed and bounded subspace will be complete and totally complete, thus compact. (A proof of) Heine-Borel theorem essentially says both are true for $X=\mathbb R^n$.

Now I would like to look at both properties individually. Clearly, (1) is equivalent to $X$ being complete: it is well-known that closed subspaces of complete space is complete, and conversely every metric space is a closed subspace of itself. However, (2) is strictly weaker than totally bounded: indeed, every subspace of totally bounded space is totally bounded, but (2) also holds for $\mathbb R^n$ which is not even bounded.

So I'm wondering: are spaces satisfying (2) being studied? What are some examples (other than $\mathbb R^n$ or its subspaces) that satisfy (2) but are not totally bounded (preferably not bounded)?

Tesla Daybreak
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  • See the following link for some information: https://math.stackexchange.com/questions/2254125/what-makes-a-metric-space-have-the-property-bounded-set-is-totally-bounded?rq=1 – Kavi Rama Murthy Jul 16 '21 at 06:31
  • $\mathbb Q$ is a 'bad' space with property (2). Any subset of a metric space wirh Heine-Borel property satisfies (2). – Kavi Rama Murthy Jul 16 '21 at 06:32
  • The linked question indeed looks very similar to what I wanted to ask. Also, thanks for pointing out (2) is closed under taking subspaces. – Tesla Daybreak Jul 16 '21 at 06:40
  • Here's a paper (a pdf) that refers to this property as the Heine-Borel property. I'm not sure how widely-known this particular terminology is, but I'd certainly say it's the most natural terminology I can think of for this property. The paper shows that a topological space is metrisable with a metric with the Heine-Borel property if the space is $\sigma$-compact, locally compact, and metrisable. The converse is clearly true as well. – Theo Bendit Jul 16 '21 at 07:01
  • Thanks, Theo, for linking the paper. – Tesla Daybreak Jul 17 '21 at 08:01
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    No worries! Just FYI, if you want to alert someone that they have a reply, put in an @ tag. – Theo Bendit Jul 17 '21 at 18:24

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