Talk:Heine-Borel Theorem

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 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 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)