Definition talk:Compact Space/Topology/Subspace

Does anyone agree that this should instead be called "compact set", as definition 2 makes no reference to the subspace topology? --abcxyz (talk) 19:04, 27 November 2012 (UTC)

We already have a redirect from Definition:Compact Set (Topology). I disagree with a rename because it's only definition 2 that makes no reference to the subspace topology. --prime mover (talk) 19:47, 27 November 2012 (UTC)

Sorry, but I still don't understand. As one need not consider the topological subspace $T_H$, I don't see why people should be bothered with the link as if it's essential to the definition.

I'm asking because of the current situation in Definition:Connected Topological Space#Set in Topological Space. I think you'll agree that these two pages should use consistent terminology. --abcxyz (talk) 16:54, 1 December 2012 (UTC)

Sorry, are we talking about "compact" or "connected" here? The two are different concepts. Besides, I'm just about to get started on giving the connectedness field a rework anyway, as I said earlier - hold your horses, I'll get round to it in a bit. I've had other things to do this last few days (both personally and professionally) and my time for ProofWiki has been limited. --prime mover (talk) 20:48, 1 December 2012 (UTC)

I'm talking about compact subspace vs. compact set. The purpose of my mentioning the "connected" page was to just to make it clear that there is a reason I'm bringing this up. --abcxyz (talk) 21:48, 1 December 2012 (UTC)

I'm sorry, I don't see the connection. What is the connection (no pun intended) between what it says on the "connected" page with what it says on the "compact" page? What is this reason? I can't see it. --prime mover (talk) 22:31, 1 December 2012 (UTC)

Both concepts do not require, but can be defined in terms of, the corresponding topological subspace.

As I suggested before, my thinking is that if a particular definition does not use the topological subspace, the link should not be put in the beginning of the definition, as it can potentially leave people wondering ("where is that used?").

In short, I think it should more be like Definition:Connected Set (Topology). Doing so also provides consistency. Comments? --abcxyz (talk) 18:27, 16 December 2012 (UTC)

I still can't understand what you're concerned about. --prime mover (talk) 22:05, 16 December 2012 (UTC)

(This was raised again at Definition talk:Proper Mapping.) In Sutherland, the only reference given, this is called a compact subset. Indeed that is the terminology that is widely used. (In fact, Sutherland writes "compact subspace" only once in the entire book.) The argument

but a set has no structure, so should not be called compact

breaks because, more generally, a red herring need not be defined as a herring that is red. Conclusions:

1. We need a definition for "compact subset" (Sutherland's, def 2 here), because it is a valid existing concept.
2. We may keep the definition for "compact subspace" (is there any source that defines it like this?). Though in this case the herring is red, so the additional definition may cause confusion.

If we do not keep the def2 of "compact subspace" (and only use it for "compact subset" instead), a thm "Subset is Compact iff Corresponding Subspace is Compact" can replace the equiv of defs. IMO, that way makes everything most transparent. --barto (talk) 06:28, 10 September 2017 (EDT)

In fact, we can even consider splitting off Definition:Compact Subset from Definition:Compact Space. --barto (talk) 06:33, 10 September 2017 (EDT)

More data from the books I have access to:

• Kelley (gen. top.) only defines "compact space" and never writes "compact subspace", only "subset" and "set".
• Sutherland (intro to met. and top. sp.) only defines "compact space" and "compact subset"; writes "compact subspace" only 2 times.

  Defines "compact space" as a special case of "compact subset".

• Mendelson (intro to top.) only defines "compact space" and "compact subset"; writes "compact subspace" only 3 times.

  Defines "compact subset" as a special case of "compact space" (for the relative topology).

• Williard (gen. top.) and Steen & Seebach (count. in top.) only define "compact space" and then use related terminology without defining it.
• Munkres (top.) only defines "compact space" and uses mostly "compact subspace", rarely "compact set" or "compact subset".

As you can see, combining Sutherland and Mendelson gives a circular def, so an intelligent choice has to be made. Seperating "compact subset" from "compact space" and treating any connections as theorems is a good option because it's less confusing and cleanest to integrate into ProofWiki. --barto (talk) 07:30, 10 September 2017 (EDT)

Lastly, note that none of them defines "compact subspace" (and all of them define "compact space"). --barto (talk) 07:32, 10 September 2017 (EDT)

"Compact subset" is as meaningless as referring to a "noisy yellow". A subset is a collection of elements that belong to a set, and there is no structure defined. "Compact" can only be defined on a set which has a topology imposed. Thus a "compact subset" makes sense only in the sense that the subset has a topology. Which topology? In the context, the subspace topology. This is important in the case where $S$ may have more than one topology imposed upon it, in which case to talk of a "compact subset" you need to know which topology is being referred to.

Hence while "compact subset" is a glossing-over of the structure of the nature of the topological space in which it is embedded. It is commonplace to talk about it, but misleading and inaccurate. --prime mover (talk) 07:54, 10 September 2017 (EDT)

Thanks for sharing your opinion, but there's one mistake though: "Compact" can only be defined on a set which has a topology imposed. As you know, it only has to be a subset of a topological space, and we don't need the relative topology. --barto (talk) 08:18, 10 September 2017 (EDT)

Oh, I just realized that the confusion arose because the definition in Sutherland has been misunderstood. I will correct his definition at Definition:Compact Space/Topology/Subspace/Definition 2 to what it should be. --barto (talk) 08:30, 10 September 2017 (EDT)

Oooh, I was looking at the second edition. First edition is much less accurate on compactness. --barto (talk) 08:43, 10 September 2017 (EDT)

You go ahead and rewrite the entirety of topology to suit yourself then, I'm done arguing. --prime mover (talk) 11:49, 10 September 2017 (EDT)

Pff, so much unnecessary discussion. The thing is that many don't bother with such fine distinctions. I think there is room for only one of the concepts "subset" and "subspace" because their difference is too subtle to be acknowledged in anything more than an "Also defined as". Furthermore the definition speaks of a subset $H$ as being "compact in $T$", so with reference to the full ambient topological space.

Furthermore, in practical situations, one is prone to drop the distinction between set and topological space to begin with, so that "$A$ is compact" can in practically all cases be understood to be covered by both variants.

This fruitless reiteration of viewpoints to the border of conflict, without an attempt to come closer together, is contrary to the goal that we all have with ProofWiki.

The only remaining point is to decide which name to choose. With appropriate redirects, this is boiling down to nothing but personal preference and overall consistence. In this case, I would say that the argument of not having to waste precious time on a disputed name refactoring tips the balance in favour of retaining the existing structure. — Lord_Farin (talk) 15:42, 10 September 2017 (EDT)

All fine for me. We could also define "subset" and "subspace" at the same time, like:

The subset $H$ is said te be compact, and the corresponding subspace $(H,\tau_H)$ is said to be compact iff [the two defs]

and add a note that this means that, by definition, the two mean the same thing. --barto (talk) 16:01, 10 September 2017 (EDT)

I reiterate, you can't have a compact subset. You can call a "compact subspace" a "compact subset" if you want to, but it won't be a "compact subset" without it being a "compact subspace". It doesn't matter what bletheringly stupid inconsistencies a lot of senile old duffers vomit out into their "textbooks", a "compact subset" is utterly stupid and meaningless, and every time I see an isntance of it I will reverse it out with extreme prejudice, and in extreme cases will delete the page. --prime mover (talk) 18:13, 10 September 2017 (EDT)

In case anybody is still interested, I have attempted to clarify my position in the "Also see" section, but I am uncertain as to whether I have done an adequate job on it. --prime mover (talk) 03:17, 12 January 2018 (EST)

Yes, looks good to me. --barto (talk) (contribs) 03:19, 12 January 2018 (EST)