Talk:Heine-Borel Theorem

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Heine-Borel Theorem vs Property

What is the exact statement of Heine-Borel theorem? I understand that the original involves only Euclidean space. For more abstract spaces not all spaces have closed bounded subsets that are compact. Is this simply a way to avoid having "Heine-Borrel Theorem" and "Heine-Borrel Property" as separate pages because their content is in essence the same? --Julius (talk) 20:46, 12 November 2020 (UTC)

The Heine-Borel Theorem states: a subset of a space (? whatever properties) is closed and bounded iff it is compact. The problems arise when you ask the questions: what does closed mean? What does bounded mean? What does compact mean? And of course "boundedness" is relevant only when you have a space on which the concept of "distance" is defined, however that is.
The motivation for this was in early topology where the brains were thinking: what aspects of the real numbers can be extracted, generalised and relaxed? That is, what is it about the real numbers that makes them special? What actually are their properties?
Hence all these "generalisations" of certain properties of subsets of the real numbers (boundedness, closedness, compactness) were found to be applicable in these more generalised spaces. What *is* the most "generalised" version of the real numbers that can sustain the structures with these definable properties such that the Heine-Borel Theorem actually makes sense? As it is established that H-BT holds for a "normed vector space", it guarantees that it will hold in *any* of the more specialised spaces which are also "normed vector spaces", that is, metric spaces, real numbers, cartesian spaces of real numbers, the whole bit.
The challenge, which needs to be stressed, is to define "closedness", "compactness" and "boundedness" in all these spaces which are "normed vector spaces" -- or rather, to demonstrate that the more-or-less intuitively defined instances of these concepts in, say, the Real Numbers, are precisely equivalent to these concepts as they are defined in a "normed vector space".
This is on a par with how we proved the "open" and "closed" as defined on $\R$ is "the same" as is defined on a metric space, and again "the same" as how defined in a general topological space -- and hence the definition of continuity. This is just another (more complicated to grasp) instance of that process. We need to show that all these instances of the H-BT are not "this is the H-BT for the reals, this is the H-BT for the cartesian space, this is the H-BT for metric spaces, ..." They are all the same thing. Whatever we do with this, we cannot lose that nugget. --prime mover (talk) 21:28, 12 November 2020 (UTC)
Ok, that makes sense. Just note that metric space is not a specialised case of normed vector space. It is the other way around. All norms induce a metric, not all metrics are induced by a norm, e.g. discrete metric does not satisfy the positive homogeneity property.--Julius (talk) 08:49, 13 November 2020 (UTC)
However it works. I haven't studied it. We have the proof for metric spaces, we show that boundedness, compactness and closure are equivalent to their metric space definitions in a NVS and the job is done. --prime mover (talk) 12:25, 13 November 2020 (UTC)

I think Julius was saying that we could rephrase this as, for example, "the real numbers have the Heine-Borel property". We say that a metric space $\struct {X, d}$ has the Heine-Borel property if a set is compact if and only if it is closed and bounded. There is a precise characterisation of such metric spaces, and there are plenty of non-trivial examples. You can also talk about the Heine-Borel property for a topological vector space $\struct {X, \tau}$, again defining this as the space having the property that a subset is compact if and only if it is closed and von Neumann-bounded. The locally convex space of smooth functions on $\Omega \subseteq \R$ is an example of such a space.

I probably will put up the Heine-Borel property at some point, and I will basically keep the theorem statements as they are, but append "That is, the real numbers have the Heine-Borel property". Caliburn (talk) 07:39, 15 June 2023 (UTC)