# Talk:Compact space

Template:Vital article Template:Maths rating

## First Paragraph

I'm physicist so be patient. That paragraph is intended as an intuitive approach so a non-mathematician should understand it, but I don't. This are the important parts (copied and pasted) '...whenever one takes an infinite number of "steps" in the space, eventually one must get arbitrarily close to some other point of the space. Thus a closed and bounded subset... of a Euclidean space is compact because ultimately one's steps are forced to "bunch up" near a point of the set..., whereas Euclidean space itself is not compact because one can take infinitely many equal steps in any given direction without ever getting very close to any other point of the space.' First, if those "steps" has to be equal you should mention it at the beginning . Second, if you are in a point and you take infinite steps towards another point (which I guess is not the infinite itself) how can you not get to that second point? Paranoidhuman (talk) 21:49, 1 April 2010 (UTC)

- I agree. This intuitive approach, as written, is neither very helpful nor rigorous. Someone needs to rewrite it, or possibly just delete it entirely. --seberle (talk) 22:27, 22 August 2010 (UTC)
- Flash out the definition first. Spend a sentence or two on intuition, if there is any. Wikipedia should be concise: fast for retrieving information. The first paragraph is chatty and misleading and should be deleted. — Preceding unsigned comment added by 50.129.229.145 (talk) 11:26, 11 August 2012 (UTC)
- Also, it's an intuitive description of "limit point compactness", which isn't the same thing as compactness in non-metric spaces.--DealPete (talk) 20:20, 31 October 2010 (UTC)
- As the lead already amply say. Sławomir Biały (talk) 22:16, 31 October 2010 (UTC)
- I do not agree. I am studying physics and I keep coming across concepts like Distributions and generalized functions which I never encountered before. It is a good thing to provide an intuitive introduction for mathematical concepts. The limitations of intuitive mode pf thinking can be stated in the article and a more rigorous treatment can be provided subsequently. There are many of us who use Wikipedia as our starting point in the (self)study of physics and mathematics and it has been very helpful. I think intuitiveness is being opposed only for 'ideological' reasons. —Preceding unsigned comment added by 210.212.179.137 (talk) 09:58, 23 January 2011 (UTC)

- As the lead already amply say. Sławomir Biały (talk) 22:16, 31 October 2010 (UTC)

The fact the opening paragraph does not feature the phrase "Every open cover has a finite subcover" is surely a weakness? 130.88.174.147 (talk) 22:20, 1 February 2011 (UTC)

- By which I mean: compactness is a topological concept. Suck it up. 130.88.174.147 (talk) 22:22, 1 February 2011 (UTC)
- We don't customarily start encyclopedia articles with abstract definitions. This definition is notoriously difficult to explain to a general audience, and so the first paragraph discusses sequential compactness (historically the first, and also the simplest notion). A formal definition appears later on, in the appropriately labeled section. Sławomir Biały (talk) 22:27, 1 February 2011 (UTC)

I've tried to explain to myself the notion of compactness by saying that a "compact" area on a flat surface could be completely filled by a finite number of arbitrary splats of paint. This is possible only if you allow the paint to touch the boundaries of your area. Likewise, a finite number of arbitrary line segments could obscure a compact line, which is only possible if you allow the pen to touch the ends of the line. Sparaig2 (talk) 06:03, 21 May 2011 (UTC)

I am a computer science student in the second semester and I must say "wow". This was one of the best first paragraphs I have ever read on a mathematical topic.--129.132.45.232 (talk) 10:08, 14 July 2012 (UTC)

- I too think the first parts of the article are among the best I have seen, but the first statement threw me off:
*a compact space is a mathematical space in which any infinite collection of points sampled from the space must — as a set — be arbitrarily close to some point of the space.*

- On first reading it sounds like there is some fixed point where all infinite sets come arbitrarily close. (This ambiguity is hard to avoid in English, or in any language for that matter, unless resorting to the kind of formulations that put non-mathematical readers off, e.g. "for every X there is Y such that...") On second reading I thought it said that for every infinite set there would be some (at least one) point in the space arbitrarily close to the set [which is trivial as every point is arbitrarily close to itself, and points in the set are also in the space]. The expression "as a set" in the present lead did not work for me. Only after reading the formal definition I could see what the statement was saying.
- Now, to make the statement intuitive, some rigor must be sacrificed, but how? I would perhaps prefer something like
*... any infinite collection of points sampled from the space must contain some "cluster"*,

- where "cluster", with quotes, is meant to be an intuitive, possibly non-standard concept. What I actually mean by a cluster, you may imagine, is that what makes a cluster point into a cluster point - sans the cluster point itself: e.g. an infinite sequence of infinite subsets of the set, such that there is nearby some point of the space not necessarily in the set, whose neighborhoods all contain a "tail" of that sequence of subsets. (And by "tail" I mean a sequence except for a finite initial sequence.) But I hope that it would not be necessary to spell it out, the statement would lead the imagination of the reader in the right direction. (If the word cluster has established meanings that conflict with this plan, perhaps "cluster of points", "clustering of points", "concentration", "accumulation", "density peak" or some other word could do the job. I agree that the lead should accommodate the largest possible set of readers. Those needing rigor can find it in the first section below the index, in the definition. Most math articles are like that.Cacadril (talk) 15:21, 7 October 2012 (UTC)
- Point above remedied? Instead of "in which any infinite collection of points sampled from the space must — as a set — be arbitrarily close to some point of the space", why not "in which for any infinite collection of points sampled from the space, there must be some point of the space to which these points are arbitrarily close"? (i.e just disambiguate the scope ambiguity!). 82.46.70.132 (talk) 05:03, 14 March 2013 (UTC)

I often turn to Wikipedia for help with mathematics and most of the articles are excellent, but in my opinion this one gets off to a terrible start, and it is difficult to recover from any enterprise if one begins with a really bad start. I understand that compactness may be a difficult concept and that one might want to start with an intuitive idea rather than a formal definition. But that is no excuse for writing things that are just plain wrong. The very first sentence is extremely bad. A "compact space" is a topological space - describing it as a "mathematical space" is vague and confusing, because in mathematics, "space" has many meanings. The link for the word "space" leads to an article for topological spaces, but the reader won't know this unless s/he follows the link. The idea described in the first sentence is sequential compactness. This is related to but not synonomous with compactness. Mathematicians define compactness using open covers. If the authors of the article want to define or describe sequential compactness first, they should explicitly state that they are doing so. Although the idea of sequential compactness seems easier to explain than the open-cover idea, the first sentence gets it wrong by suggesting that the entire sequence must converge, and not just a subsequence. It appears the author(s) are trying to avoid the idea of subsequences so early in the article. If this is an issue, maybe one should provide an example of a sequence in [0,1] and a convergent subsequence, to illustrate the meaning of "subsequence". Avoiding the ideas of subsequence and convergence leads to such absurdities as "...some of those points must get arbitrarily close to some real number in that space..." How can a point get arbitrarily close to a real number? Compactness simply is not an easy subject, and it may be a mistake to try to avoid technical terms altogether. Anyone who is reading this article in the first place probably already enough mathematics background to understand a good definition of compactness. From the other comments on this page, it appears that the attempt to be informal has made the article more rather than less confusing. The second sentence of "Introduction" section continues the error of confounding compactness with sequential compactness. A few paragraphs down, it is suggested that a subset of Euclidean space is defined to be compact if it is closed and bounded. This is not a definition, this is the result of a theorem (the Heine-Borel theorem). I didn't read any farther in the article. I agree with a comment that there should not be separate articles for compact space and compact set. If they were merged into one, it might make it easier for readers to understand both.

I don't mean to offend whoever wrote this article, and I know it is easy to criticize. Whoever wrote it probably knows more than I do about topology, and it appears to contain a lot of valuable information, once one gets past the definition. But I think that in mathematics, there is no excuse for writing something that is imprecise or incorrect, even if the goal is to make something easier to understand. In this case, I don't think that goal was achieved.Gsspradlin (talk) 23:10, 29 October 2013 (UTC)

- Formal details can be found in the "Definition" section for those who have enough mathematical background to understand the open cover definition of compactness. The lead is meant to be readable by a broad audience, and it is not incorrect as currently written. It is already asserted that there are different, inequivalent, notions of compactness and that the version described in the first sentence is sequential compactness. Sławomir Biały (talk) 01:30, 30 October 2013 (UTC)

On the contrary, the very first sentence has at least two errors. It defines compactness incorrectly as sequential compactness, and in the attempt to define sequential compactness, it is suggested that the entire sequence, not just a subsequence, must converge to a point. If you write something later like "Oh yeah, that definition was actually sequential compactness", you have still started with a bad definition. If you want to convey the "flavor" of a compact set to people with little mathematical background, then why not give some properties and examples, such as finite sets, close and bounded subsets of Euclidean space, and give the fact that a continuous real-valued function on a compact space has a maximum and minimum value. That fact may be the reason why any non-mathematicians are interested in compact spaces/sets at all. Then give a real, correct, definition of compact space for those who can handle it. I don't think the open-cover definition is really any more difficult than the sequential-compactness definition. One must know what an open set is to understand the former, and what a subsequence is to understand the latter. If a reader doesn't know either, then you can't expect them to understand a definition of either. You wouldn't try to define, say, the convergence of an infinite series to someone who can't compute 2 + 3. This article seems to attempt to define compact spaces without requiring the reader to know anything at all about topology, even topology of metric spaces. I think that's a bad idea. Gsspradlin (talk) 19:56, 30 October 2013 (UTC)

- Sequential compactness is among several different types of compactness, and was actually historically the first to be introduced by mathematicians. In fact, as late as the first few decades of the 20th century "compact" meant "sequentially compact". This is discussed in both the lead and the article body. There is absolutely nothing wrong with presenting this notion of compactness first, as long as the article indicates clearly that it is about sequential compactness. The reason that the authors of the article decided to present the material this way is that the notion of sequential compactness is much easier to explain in a manner that is accessible to as wide an audience as possible (see WP:MTAA and WP:LEAD). Commentators on this discussion page have included readers from a wide variety of backgrounds, not all of whom would be comfortable with the notion of compactness defined in terms of covers (though that more usual form of compactness is also discussed in the lead of the article). In short, it is simply wrong to declare that "sequential compactness" is not "compactness": it is one among many definitions of compactness that mathematicians use that are equivalent in good cases, but inequivalent in general.
- Also, contrary to your statement, the lead does not suggest that the sequence itself must converge. (This was the subject of a recent discussion on this very page, where an amenable wording was reached in fact.) Moreover, the first example given in the lead is
*not*itself a convergent sequence: it has two convergent subsequences that converge to different limits. - Finally, both the lead and the article itself already do convey the "flavor" as you suggest, by including a discussion of closed and bounded subsets of Euclidean space. Sławomir Biały (talk) 20:48, 30 October 2013 (UTC)
- But the default meaning of
*compact*in a modern mathematical context is not in fact sequential compactness; it's "every open cover has a finite subcover". I agree that the lead paragraph should be as accessible as possible, but it should not mislead, and as it stands it is misleading. With the current text, there's a very real risk that a naive reader could read the first sentence, skim the rest, and come to the conclusion that compactness is generally defined as sequential compactness, which is not true. - I might be OK with starting with sequential compactness provided it's done in such a way that you can't miss that it's not the general definition. That would have to be made very very very explicit. I mean you really have to HIT THE READER OVER THE HEAD with it very very very hard. --Trovatore (talk) 00:13, 31 October 2013 (UTC)
- Agree. Paul August ☎ 01:52, 31 October 2013 (UTC)

- But the default meaning of

### How about starting with closed and bounded in R^n?

The problem with the existing text is that it suggests (no, actually, it flat out asserts) that a compact set is a sequentially compact one, which is just simply not standard mathematical usage. Sławomir's justification is that it is easier to explain, which may be true, but while we want to be as accessible as possible, we must never indulge in lies to children; I hope everyone agrees that those are abominable and beyond the pale.

We could try to find wording that would start with sequential compactness but make it more explicit that it's not the general definition, but it's going to be awkward.

So how about this? We start with an *even simpler-to-explain* definition, that of being closed and bounded, but specify that the real concept is a generalization thereof. Just off the top of my head; this can no doubt be much improved:

- In the mathematical discipline of general topology,
**compactness**is a property that generalizes the notion of a subset of Euclidean space being closed (that is, containing all its limit points) and bounded (that is, contained within some finite cube of the appropriate dimension). This notion is generalized to more general topological spaces in various ways, including sequential compactness, which means (...). The standard meaning of the unqualified word**compact**is (here put every open cover has a finite subcover, but with appropriate explanations and glosses).

What do you think? --Trovatore (talk) 00:34, 31 October 2013 (UTC)

Would this be acceptable? I have moved your suggested edit regarding the standard unqualified use of the term "compact" to the third paragraph, where that definition is actually addressed. Sławomir Biały (talk) 01:03, 31 October 2013 (UTC)

In the mathematical discipline of general topology, compactness is a property that generalizes the notion of a subset of Euclidean space being closed (that is, containing all its limit points) and bounded (that is, contained within some finite cube of the appropriate dimension). This notion is generalized to more general topological spaces in various ways. For instance, a space is sequentially compact if any infinite sequence of points sampled from the space must eventually, infinitely often, get arbitrarily close to some point of the space. The Bolzano–Weierstrass theorem states that a subset of Euclidean space is compact in this sense if and only if it is closed and bounded. Examples include a closed interval or a rectangle. Thus if one chooses an infinite number of points in the closed unit interval, some of those points must get arbitrarily close to some real number in that space. For instance, some of the numbers 1/2, 4/5, 1/3, 5/6, 1/4, 6/7, … get arbitrarily close to 0. (Also, some get arbitrarily close to 1.) Note that the same set of points would not have, as an accumulation point, any point of the open unit interval; hence that space cannot be compact. Euclidean space itself is not compact since it is not bounded. In particular, the sequence of points 0, 1, 2, 3, … has no sub-sequence that ultimately gets arbitrarily close to any given real number.
Apart from closed and bounded subsets of Euclidean space, typical examples of compact spaces include spaces consisting not of geometrical points but of functions. The term Various equivalent notions of compactness, including sequential compactness and limit point compactness, can be developed in general metric spaces. In general topological spaces, however, the different notions of compactness are not necessarily equivalent, and the most useful notion, introduced by Pavel Alexandrov and Pavel Urysohn in 1929, involves the existence of certain finite families of open sets that "cover" the space in the sense that each point of the space must lie in some set contained in the family. The standard unqualified use of the term |

Hmm. It's not too bad. Certainly I think it's a huge improvement over the existing text. My only worry is that the standard definition is maybe a little hard to find, for people who are actually looking for it. --Trovatore (talk) 01:06, 31 October 2013 (UTC)

I agree that the proposed change (what is in the box) is a huge improvement over what is there now. As several people have pointed out, the current article starts with a false assertion because it attempts to give an easy definition for a concept that is not easy (though not terribly difficult). It's better to begin with a description of the idea or flavor of compactness, and then give a good definition. Gsspradlin (talk) 01:37, 31 October 2013 (UTC)

- Since there have been no strong objections so far, and the consensus seems to be that this is an improvement, I have gone ahead and pasted this in. Sławomir Biały (talk) 11:29, 31 October 2013 (UTC)

- Suggestion: early use of a second style of example, closed manifolds. Naturally after Whitney we know that these examples are not more general than compact subsets of Euclidean space. Less general, more geometric, can have pictures. Charles Matthews (talk) 08:15, 8 November 2013 (UTC)

## Examples for spaces which are NOT compact

This article lacks examples of spaces which are not compact, which in my opinion are important to the understanding of the definition. I would suggest adding some simple examples (maybe with an explanation or a proof) - such as the open unit interval, the real line, and C(0,1).

—The preceding unsigned comment was added by 132.69.230.37 (talk) 07:58, 26 March 2007 (UTC).

- Well in the first few lines it says that in Euclidean space a subset is compact is it is closed and bounded. So for a counter example chose a subset that is not closed, say , one that is not bounded, say , or one that is neither closed nor bounded, say itself. Dharma6662000 (talk) 01:22, 26 August 2008 (UTC)
**R**is closed in**R**. In any case, I don't think there's much point giving so many examples of non-compact subsets of**R**^{n}when we already have two in the lead, plus the full characterization of compact Euclidean sets in its own section. We could maybe use some more interesting examples of noncompactness in the article though (there's already two hidden at the bottom of the examples list). Algebraist 01:31, 26 August 2008 (UTC)- Yes, it is absolutely true that
**R**is closed, because it clearly contains all of its limit points. A good example of a set that is not compact is a bounded set of the rational numbers**Q**. It is not compact because it is not closed. I also cannot understand people who belly-ache about "too many examples" - because as a math student in college-level and graduate-level courses, examples was what "brought the subject to life". For those of you who subsist on abstract thinking all the time - maybe go reside with Osama bin Laden?98.67.103.56 (talk) 18:15, 8 August 2009 (UTC)

- Yes, it is absolutely true that

- As with my above example of an intuitive way of looking at compact sets, the alternate to a compact set is a set that can't be covered by a finite number of splats of paint, simply because you aren't allowed to touch the boundaries. Because of that, there is always a gap that isn't covered, no matter how many times you add a splat. This seems pretty darned intuitive. Is there something I am missing?Sparaig2 (talk) —Preceding undated comment added 03:27, 23 May 2011 (UTC).

## Notions

Zundark, sorry that I screwed up the implications between the various compactness notions. --AxelBoldt

That's okay. I should have noticed it at the time. --Zundark, 2001 Dec 15

I'm new to Wikipedia, so I don't know if this is a good place to present a question. Prove or disprove: There exists a compact space X which can be covered by two (intersecting) open sets U and V so that no two compact sets K and F, with K a subset of U and F a subset of V, cover X. The example, if standard set theory can provide one, will be more complex than it seems at first glance. This is connected with properties of the compact-open topology. --Roman.

- For a non-hausdorff space, it's easy. Just take an indecomposable space (e.g., the reals with the Zariski topology). For a hausdorff space, it's impossible. Apply normality to the complements of U and V, then take the complements of your resulting open sets. These are K and F. —Preceding unsigned comment added by 24.240.36.203 (talk) 21:02, 11 May 2010 (UTC)

## brackets

Why is it important to nowiki the half-open interval, but leave the closed interval as is??? Revolver 11:46, 9 Nov 2004 (UTC)

## help

Can someone help me understand this:

"The modern general definition calls a topological space compact if...any collection of open sets whose union is the whole space has a finite subcollection whose union is still the whole space."

Does [0, 1] fit the definition of a compact space simply because there exist no collections of open sets whose union is [0, 1]? At least I cannot think of any--if you can think of some, please help me. Or should the word "is" be replaced by "contains"?

- There are collections of open sets whose union is [0,1]. For example, { [0,.6), (.5,1] }. Remember that the open sets of [0,1] are the sets of the form [0,1]∩
*U*, where*U*is an open set of**R**. --Zundark 07:32, 12 May 2005 (UTC)

- I'm confused about this one too: taking the link open set's under metric space (
**R**is a metric and topological space), we see that: A subset U of a metric space (M,d) is called open if, given any point x in U, there exists a real number ε > 0 such that, given any point y in M with d(x,y) < ε, y also belongs to U. But the '1' in (.5,1] break that statement, no ? - Update: Ok, so the key is relative topology. I think someone should point that out in compact space, cover and possibly open set. The way it is now (4th August, 2005), we can deduce that a subset of a topological space is compact if it has a finite open subcover of its open cover. Then if you look up (open) cover, you conclude that a cover (of a subset S of a topology X) is a union of *open sets* in S which gives (hence the name "cover") the subset. Now, 'open' might be taken (which would be incorrect) to be from the original topology (and not the subspace one).

- I'm confused about this one too: taking the link open set's under metric space (

- Neither [0, 0.6) nor (0.5, 1] are open sets, due to the elements 0 and 1 respectively. Also, the word "is" should indeed be replaced by "contains", as per the definition of "open cover." Hence, {(-0.1, 0.6), (0.5, 1.1)} would suffice as an open cover of [0,1] which has a finite subcover of [0,1]. --Tekhnofiend (talk) 22:15, 10 January 2010 (UTC)

- Nevermind, I think I've confused the discussion to be about subsets of topological spaces rather than the spaces themselves. You should disregard my earlier comment. --Tekhnofiend (talk) 23:50, 10 January 2010 (UTC)

## merge with Compact set proposal

It seems that compact set and compact space are rather duplicate. Shouldn't we merge these articles, taking care that both "unbounded and closed" and "finite subcover" are well treated and keeping the relation between the different definitions clear? I'm willing to do the merge myself, but comments/suggestions/objections are welcome. --Lenthe 11:29, 27 July 2005 (UTC)

- You are right. It would however be preferable if compact set were merged into compact space, in such a way as to preserve the elementary flavour. That is, to be friendly first to those who need the idea on the real line, rather than the generality of topological spaces. Charles Matthews 12:04, 27 July 2005 (UTC)

- I've been meaning to do this merge for a long time, but have never gotten around to it. Actually, it looks like much of the material in compact set belongs in the article on the Heine-Borel theorem. The rest of it should go in compact space. -- Fropuff 13:46, 27 July 2005 (UTC)

- OK, I've moved the content of compact set (now a redirect) to compact space and Heine-Borel theorem and started polishing both articles a bit. Help on polishing is very welcome. --Lenthe 14:59, 3 August 2005 (UTC)

- For the elucidation of foreigners and everyone else, "OK", is always written with All Capitals. The best that historians can deduce, it is an abbreviation for "Old Kinderhook", a nickname that President Martin Van Buren of the United States used.98.67.103.56 (talk) 18:21, 8 August 2009 (UTC)

## Name Change Proposal

It might be more appropriate to name this page "compactness", especially considering the "history" section. It isn't a history of compact spaces, but rather of compactness. It seems a more intuitive name as well. Fell Collar 01:51, 2 March 2006 (UTC)

- Yet "compactness" can refer to a large number of things (even in mathematics), while "compact space" relates more clearly to the topological property. --Lenthe 08:40, 2 March 2006 (UTC)
- What other things are you thinking of? Fell Collar 18:11, 2 March 2006 (UTC)
- Nobody ever answered this question. "Compactness" refers to automobiles, pistols, portable radios, the nuclear reactors in submarines, the small varieties of wrestlers and boxers, white dwarf and black dwarf starts, and 101 other things.98.67.103.56 (talk) 18:29, 8 August 2009 (UTC)

- At any rate, the article isn't just about compact spaces. It's also about compact subsets of metric spaces or
**R**^{n}. Moreover, while "compactness" may refer to a number a things in mathematics, it seems to me that this is by far the most common. To clear up any confusion, we could always add a disambiguation link at the top of the article. --Fell Collar 15:40, 9 March 2006 (UTC)- First of all, didn't we just finish merging compact set into this article? It almost sounds like you want to split again (which I don't support). Second of all, the definition of compact sets in
**R**^{n}or metric spaces is simply that they are compact spaces in their subspace topologies. So the name "compact space" covers everything. I do not support a name change. -lethe^{talk +}16:43, 9 March 2006 (UTC)- I didn't mean to suggest we split the article again; I was suggesting that we keep all the content here, but change the name to something more inclusive of perspectives outside of topology that would justify the section about history as well. Compactness is also important in analysis, but one rarely refers to "compact spaces" there, nor does one reference topology in the definition of compact (Wheeden and Zygmund's "Measure and Integral", for example, refers only to "compact sets" and gives the finite open covering definition). This is a valid perspective as well. Rather than pick one point of view over the other, I'm suggesting we simply name the article "compactness" and list both perspectives. This seems rational to me, and in keeping with Wikipedia's NPOV guidelines. --Fell Collar 19:08, 9 March 2006 (UTC)

- First of all, didn't we just finish merging compact set into this article? It almost sounds like you want to split again (which I don't support). Second of all, the definition of compact sets in

- What other things are you thinking of? Fell Collar 18:11, 2 March 2006 (UTC)

The appropriate name for this page is **compact space**. Please don't rename it. If you want to start a new article on compactness which discusses the various forms of compactness (sequential, countable, locally, para-) and their histories that would be fine. You could move some of the material from this page there. -- Fropuff 05:08, 10 March 2006 (UTC)

- I'd be okay with that, if other people are as well. I expect there would be a considerable amount of redundancy with this article, which might be a concern given the recent merge. What do other people think? --Fell Collar 19:58, 10 March 2006 (UTC)
- The only reason I would think that a separate page was needed would be if this page were too long. This page is currently not very long, so I do not support such a split, though I would change my mind if someone showed up who wanted to write all the stuff about those other forms of compactness which would make this article too long. Are you that person? -lethe
^{talk +}22:59, 10 March 2006 (UTC)- I'm not particularly interested in writing all that, and I don't want this to be any more controversial than it already has been, so I'm just going to leave it alone. --Fell Collar 03:55, 11 March 2006 (UTC)

- The only reason I would think that a separate page was needed would be if this page were too long. This page is currently not very long, so I do not support such a split, though I would change my mind if someone showed up who wanted to write all the stuff about those other forms of compactness which would make this article too long. Are you that person? -lethe

## Mnemonical rule to remember Compact space

This rule came from Russia in a form of a joke: Mathematician is talking to a pretty girl: - You are so compact... Girl fondly specified his answer: - Do you mean well-shaped and thin? - No. Closed and bounded! --Yuriy Lapitskiy 22:26, 9 March 2006 (UTC) hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh

## Compactness vs. completeness

(Copied from user talk:AxelBoldt:) Do you happen to know the proof (or where it can be found) of the statement

A metric space *X* is compact if and only if every metric space homeomorphic to *X* is complete.

that you added to the compact space entry in 2002? Slawekk 17:41, 26 May 2006 (UTC)

I'm not so sure anymore that the statement is correct and I have removed it for now. The one direction is clear: if *X* is compact, then every space homoeomorphic to *X* is compact, and every compact metric space is complete. For the other direction, I wanted to use the Stone–Čech compactification β*X* of *X*: if β*X* is metrizable, then the subspace *X* of β*X* is a metric space homeomorphic to *X* and is therefore complete by assumption, and a complete subspace of a compact metric space is itself compact. Problem is, I'm not sure whether β*X* is metrizable. So I'm missing the following statement:

- The Stone–Čech compactification of every complete metric space is metrizable.

AxelBoldt 21:16, 26 May 2006 (UTC)

I don't think the missing statement is true. *X* has a metrizable compactification iff *X* is 2nd countable and tychonoff [1]. Of course not every complete metric space is second countable.
Slawekk 22:24, 26 May 2006 (UTC)

- You are right, thanks a lot! AxelBoldt 23:54, 27 May 2006 (UTC)

## Compact space vs. complete algebraic variety

An algebraic variety is complete iff (by definition):

for all algebraic variety , the second projection is a closed map.

Is something like this true for topological spaces? (According to Dieudonne this should be true) That is: Let be a topological space. Then

is compact iff (for all topological space , the second projection is a closed map ).

Could someone help me? Thanks in advance. —Preceding unsigned comment added by Pepe 986 (talk • contribs) 00:23, 12 March 2008 (UTC)

- Yes. A Hausdorff space
*X*is compact if and only if for every Hausdorff space*Y*, the projection*X*×*Y*→*Y*is a closed map. I believe that Hausdorffness is required. Ryan Reich (talk) 20:53, 28 April 2009 (UTC)

## What should be done here?

This page is labelled as being "barely B class" and of high importance. I don't see anything terribly wrong with it, although it could use some editing for internal consistency (a few things get mentioned several times seemingly needlessly, and there is a lack of parallelism in certain symbols and fonts used).

Of course the article does not say everything that one could possibly say about compactness (no single article could), but I wonder what people think is missing here?

One thing that springs to mind is that it would be nice to mention the compactness theorem in model theory and how it does, in fact, assert the compactness of a certain Boolean space. Plclark 10:09, 8 October 2007 (UTC)Plclark

- I think this article should give much more of an overview of the concept of compactness and how it is used in various parts of mathematics. It should also summarise the articles on the various variant definitions. There are probably plenty of other things that are missing. I am sure that an entire book can be written on compactness, and that this article has the potential to be featured. --Hans Adler (talk) 10:29, 21 April 2009 (UTC)

## compact vs. connected

can someone clarify the differences between compact & connected? 76.68.228.196 (talk) 16:29, 24 October 2008 (UTC)

- They are totally different concepts. I really can't see how you could confuse them. Algebraist 17:51, 24 October 2008 (UTC)
- See compact space and connected space. Maybe you were just looking at the wrong definitions.

Topology Expert (talk) 21:22, 7 December 2008 (UTC)

- Simple answer: The union of [0,1] and [2,3] is compact but not connected. (0,1) and [0,∞) are each connected but not compact.
- I think I have encountered this confusion before. (And personally I sometimes have problems distinguishing words that start the same.) Perhaps the article should give these simple examples. But then, perhaps it's enough to have them here on the talk page. --Hans Adler (talk) 10:25, 21 April 2009 (UTC)

- Reading the Talk page is usually my last resort to sort out an issue with the Article. I think that a note in the beginning of the Article about compact spaces and the above mentioned example in the example section should solve this issue.
- TomyDuby (talk) 16:28, 21 April 2009 (UTC)
- I am very forgetful, and I hope there is a way, when I need it, I can find it easily. The simple answer in "Compact vs. connected" section is very nice. It is much easy for readers as me to get the idea. By reading the article, I didn't get anything but headache.

## Compact vs. Tight

Is there anyone can elaborate the difference between compact vs. tightness? Jackzhp (talk) 16:50, 10 July 2010 (UTC)

## Help for Math Student

I am very confused by the definition of compactness in conjunction with the definition of Cover, and I figured that this might be something that should be explained better in the article. Either I am understanding something terribly incorrectly, and there is nothing in the article that addresses or clears my misunderstanding, or the Heine-Borel theorem is wrong and [0,1] is not compact.

If [0,1] is compact, according to the definitions on the articles I mentioned above, there should be no open cover without a finite subcover. Then is A={(-0.1, 0.5), (0.4, 1.1)} an open cover for [0,1]? If not, why not? If so, how can I make a subcover for A? This seems very similar logic used in the set (n-1,n+1) that was used to prove that there exists a cover of the integers that has no subcover. Can someone clear this up either in this article or in the article Cover (topology)?

Thanks 24.7.179.171 (talk) 01:39, 20 August 2009 (UTC)

- I do not quite understand your confusion. The cover
*A*you have given is indeed an open cover of [0, 1] (each element of [0, 1] is in one of the elements of*A*). In fact, it is a finite cover because there are only two elements in*A*. Therefore, it is a finite subcover of itself, and this does not contradict compactness of*A*. - An open cover of [0, 1] is a collection of open sets whose union is [0, 1]. To create a finite subcover, one must delete some open sets in this collection so that only finitely many are remaining, and prove that this finitely many does indeed constitute a cover. If the sets in the open cover themselves have infinitely many elements, it does not change the number of sets in the cover (in other words, the number of elements of the open cover does not relate to the number of elements of a particular open set that is one of many which constitutes the cover). The key point here is the notion of "a set of sets". When one studies sets of objects, it does not matter what the objects are, or how many elements they have. In the case of open covers, one considers a set of open sets (so the objects are open sets). With compactness, what matters is not
*what*the open sets are (as long as they constitute a cover), but*how many*there are. Hope this helps. --PST 07:32, 20 August 2009 (UTC)

- Oops! I made a stupid mistake: I forgot that a set is a subset of itself. Very silly of me. Then finite covers are trivial when discussing compactness, and one only needs to consider whether or not infinite covers have
*finite*subcovers. Thank you. 24.7.179.171 (talk) 03:38, 23 August 2009 (UTC)

- Oops! I made a stupid mistake: I forgot that a set is a subset of itself. Very silly of me. Then finite covers are trivial when discussing compactness, and one only needs to consider whether or not infinite covers have

- It's my understanding, from the definition of an open cover, that it is incorrect to say that an open cover of [0, 1] is a collection of open sets whose union is [0, 1]. By definition, an open cover of [0,1] is a collection of open setes whose union CONTAINS [0,1]; equality is not required. This confusion arose earlier in this talk page and even creeped it's way into the main article. I'll make the change in the main article. Apologies if I'm somehow mistaken, in which case let me know! --Tekhnofiend (talk) 22:15, 10 January 2010 (UTC)
- I was referring to sets open in the subspace topology on [0,1], when defining an open cover of [0,1]. --PST 02:34, 11 January 2010 (UTC)
- The definition in the article is correct for an open cover of a topological space. It doesn't say anything about the slightly different question of what an open cover of a subset of a topological space is. In fact, the article doesn't seem to cover subsets properly at all; there should probably be a note somewhere saying that a subset A of a space X is compact iff it's a compact space when endowed with the subspace topology. Algebraist 22:34, 10 January 2010 (UTC)
- I see what you mean now. Thanks for the clarification. --Tekhnofiend (talk) 23:50, 10 January 2010 (UTC)

- It's my understanding, from the definition of an open cover, that it is incorrect to say that an open cover of [0, 1] is a collection of open sets whose union is [0, 1]. By definition, an open cover of [0,1] is a collection of open setes whose union CONTAINS [0,1]; equality is not required. This confusion arose earlier in this talk page and even creeped it's way into the main article. I'll make the change in the main article. Apologies if I'm somehow mistaken, in which case let me know! --Tekhnofiend (talk) 22:15, 10 January 2010 (UTC)

## clarify

I understood this stuff 30 years ago, but I've forgotten most of it. I have a question/comment about the lead.

Suppose you have a closed disc of radius 2 centered at the origin. Suppose your infinite number of steps go (1,0), (0,1), (-1,0), (-1,-1), (1,0), etc. What point is it approaching?

Am I misunderstanding it or is the description in the lead not complete? Shouldn't it be that for every infinite series of points in the space that converges, it converges to a point in the space? Bubba73 ^{(Who's attacking me now?)}, 18:56, 16 December 2009 (UTC)

- The description in the lead is not intended to be completely rigorous. Sequential compactness, which is properly defined later on, states that any sequence whatsoever has a convergent subsequence. Obviously any convergent sequence converges (...to a point...in the space), so the alternative you offer is vacuous as stated. In spirit it is similar to that of a closed set (a subset of a metric space which contains the limits of every sequence that converges to a limit in the larger space) or a complete space (a metric space in which every Cauchy sequence converges). Both of these conditions are strictly weaker than compactness. Sławomir Biały (talk) 21:42, 16 December 2009 (UTC)
- Hey, I've come here to ask
*that*same question... ― ___A._di_M. (formerly Army1987) 21:35, 4 April 2010 (UTC)

Well, Sławomir Biały, you're right about the theorems. But although as Paranoidhuman had said, the first paragraph should be easy to understand for laymen, however, it should be relatively strict as well. I think the metaphor the article gives is definitely not clear enough and also not nature-revealing. Trvoldemort (talk) 07:09, 8 May 2010 (UTC)

- I'm certainly open to suggestions about better ways of expressing it for laymen, but this is the best I've found so far. The lead should be understandable by laymen: it does not need to transmute laymen into experts on the subject. Paranoidhuman seemed to think that the steps had to be equal (but they don't) and had some trouble with a notional paradox of infinity (Zeno's paradox). Well, we can't really do much about that. I have no idea what "theorems" you are referring to. Nowhere in this discussion page have I referred to any theorems. The above post was simply referring to different definitions of related ideas: closed set and complete space. Sławomir Biały (talk) 11:33, 8 May 2010 (UTC)

## Peanut gallery challenge

There seem to be lots of criticisms from the "peanut gallery" that the present lead fails to give a notion of compactness that is both understandable to laypersons and is completely rigorous. While that seems like something of an oxymoron, the criticism has been made—quite explicitly in recent posts. So I've kind of lost patience with the peanut gallery here, and I am about to throw the WP:SOFIXIT gauntlet. The version of compactness that, to me, seems amenable to lay understanding is that of sequential compactness. So the challenge to the hecklers here is to please explain the following in a way that satisfies all requirements for the lead:

- A space
*X*is**sequentially compact**if every sequence in*X*has a subsequence that converges to a point of*X*.

Please refrain from further unconstructive criticism (especially from folks who have not even read the article) until the above challenge is met. I, for one, think the present version of the lead explains this concisely and intuitively, but obviously I am in the minority. Sławomir Biały (talk) 15:07, 8 May 2010 (UTC)

- I heavily edited the lead's first paragraph. I decided to say the "steps" one takes have to be distinct ("one takes an infinite number of "steps" to different points in the space"), since if they're not distinct sequential convergence is trivial, and the layman's description doesn't need to split that hair. Hopefully this takes care of the "clarify" criticism above. The original also implied the sequence converges, instead of that a subsequence converges. I understand it wasn't supposed to be rigorous, but that's a pretty big mistake. I also noted right off the bat that there are several versions of compactness--I imagine people reading the page will want to know they have to figure out which version fits their needs, and they'd prefer that information before they skip the rest of the long intro. Finally, I turned the Euclidean space comments into two examples (if you have a better sequence than 1, 1/2, 1, 1/3, 1, 1/4, ..., feel free to suggest it), since a representative example is worth tons of pure exposition. It could certainly be improved, but I find it difficult to do so knowing what compactness is without having more space. 67.158.43.41 (talk) 22:26, 5 October 2010 (UTC)

- Overall, I like the edit. I don't agree that the original wording necessarily implied convergence of the sequence, but the new edit eliminates this ambiguity nicely. Also, I am not big on having detailed examples in the lead. This was the original reason that I created the "Introduction and definition" section: the lead had spun out of control with the number of examples and different issues it tried to present. However, I'll be the first to admit that the example of sequences in the unit interval should help to instill an appropriate mathematical understanding of "steps" in a space, and probably is needed. Finally, I hope there is a way to move the discussion of the other definitions of compactness later, so that the first paragraph gets directly to what a compact space is, rather than leave it hanging as an "abstract mathematical space". In fact, it's almost better to go down this route:
- a
**compact space**is an abstract mathematical space in which, intuitively, whenever one takes an infinite number of "steps" to different points in the space, some of the steps must get arbitrarily close to a particular point of the space. (More precisely, this is the description of a sequentially compact space, and there are other distinct notions of compactness covered later.)

- a
- But I feel that parenthetical comments are a mark of poor writing (which is not to say that I don't use them). Sławomir Biały (talk) 23:30, 5 October 2010 (UTC)

- Overall, I like the edit. I don't agree that the original wording necessarily implied convergence of the sequence, but the new edit eliminates this ambiguity nicely. Also, I am not big on having detailed examples in the lead. This was the original reason that I created the "Introduction and definition" section: the lead had spun out of control with the number of examples and different issues it tried to present. However, I'll be the first to admit that the example of sequences in the unit interval should help to instill an appropriate mathematical understanding of "steps" in a space, and probably is needed. Finally, I hope there is a way to move the discussion of the other definitions of compactness later, so that the first paragraph gets directly to what a compact space is, rather than leave it hanging as an "abstract mathematical space". In fact, it's almost better to go down this route:

- I prefer the first sentences the way they are, unfortunately. I certainly agree it would be nice if the article could immediately say what compactness is instead of "leaving it hanging". However, I think the most agreed-upon definition (finite subcovers of all open covers) is far too abstract to use, and it just doesn't flow well to say "here's a definition" followed immediately by a comment that says "but wait there are many others that sometimes are equivalent but sometimes aren't (and, though I'm not telling you, this probably isn't even the most common one)". Feel free to rework any parenthetical comments. If it were up to me, I would probably rework the intro quite a bit more to first clearly overview the different notions of compactness by naming them but not going into details, noting that sometimes they're equivalent and sometimes they're not. The second paragraph might then explain sequential compactness in an intuitive manner, using Bolzano-Weierstrass and the examples I've given, and more importantly noting that this formulation is equivalent to the others in metric spaces so it's not random to use it over the others. The existing 3rd paragraph would be gutted to create the first paragraph, and the existing 2nd paragraph would become the new third. But, I understand wanting to give a definition right away, so I won't make the changes without discussion. 67.158.43.41 (talk) 10:56, 6 October 2010 (UTC)

- I'd like to highlight the issues as I see them. The purpose of a lead is to give context for the topic and summarize the rest of the article in a manner that can be grasped as much as possible without a detailed knowledge of the subject. I think we're in agreement that, at least, the usual definition using open covers is simply too abstract to include in a general-purpose introduction to the subject. This belongs further down in a definition section. The current structure of the lead consists first of a paragraph saying what a compact space is in the most simple and rarefied way possible followed by a paragraph that attempts to give a summary of the "History" section, highlighting the history and importance of compactness, and finally a third paragraph giving some slightly more technical background on the usual notion of compactness, theorems, etc. This may not be the perfect structure, but I think that gutting the first paragraph to create a new first paragraph runs the risk of making the lead much more technical. It seems that the first paragraph would then need to bring in general topological spaces, metric spaces, open sets—very abstract ideas that the typical reader will have no idea about. Of course, a lot of this depends on execution as much as anything else, and so I look forward to future edits with anticipation. Sławomir Biały (talk) 11:35, 6 October 2010 (UTC)

## Lead and introduction

I think the recent edit to the lead is precisely what was needed. My first reaction was that it was too long, and that it should somehow be rearranged to push some of the details down into the "Introduction" section. But perhaps instead the lead should be left as it currently stands, modulo minor edits, and rather the first paragraphs of the introduction section should be revised. I'd like to get some input on this. Sławomir Biały (talk) 13:28, 9 January 2011 (UTC)

## Compactness (topology)

As Compactness (topology) redirects here, this article ideally needs a main section dedicated to the property of compactness, as opposed to a compact space Brad7777 (talk) 13:30, 6 February 2012 (UTC)

- I don't understand. This whole article is about compactness as it is defined in topology. A compact space is a topological space (or a subset of a given topological space in the subspace topology) with the property of compactness. Arguably, the article could be renamed from Compact space to Compactness (topology), but I see little benefit in that. — Tobias Bergemann (talk) 14:02, 6 February 2012 (UTC)

## "Smallness" Analogy

I think that describing compactness as a measure of, in some sense, "smallness", is a bit misleading to the uninformed reader, especially since compact spaces can very often be supersets of non-compact spaces (long line vs. real line, etc). Perhaps a change of wording is in order? Thoughts? — Preceding unsigned comment added by 207.239.94.2 (talk) 20:31, 23 March 2012 (UTC)

- I don't think a change in wording is needed. The lead already goes to extraordinary lengths to convey that there is no simple obvious characterization of compactness. It also says that sometimes compact spaces can be quite "large". Sławomir Biały (talk) 01:28, 24 March 2012 (UTC)

## Abominable first paragraphs!!!

The first paragraphs, supposed to introduce the reader to the concept of compactness, could hardly be worse. (At least they are grammatical.)

**No information whatsoever** is conveyed by saying that a compact space is one possessing the quality of compactness.

Also, virtually **no information whatsoever** is conveyed by saying that the concept of a compact space is hard to describe intuitively.
That very same space on the page could be used to at least **attempt** to describe the intuitive idea of a compact space. In the most common settings like Euclidean space, compactness is **not** hard to convey intuitively. That is true only in the most general settings, and **there is no need to discuss the most general settings in an introductory paragraph.**

(Here is one way to do this could be something like: "For the simplest kind of compact space -- one that is a subspace of some Euclidean space -- a compact space is one in which any infinite set of points must have a limit point." This could be followed by the essence of the Heine-Borel argument for the closed unit interval, and then even for a closed and bounded set in R^n.)

This is an encyclopedia, whose purpose is to convey information. Sentences and paragraphs that convey no information will merely frustrate the reader.Daqu (talk) 18:00, 11 September 2012 (UTC)

Couldn't agree more. 129.187.147.189 (talk) 12:20, 20 September 2012 (UTC)

## First line

"In mathematics, specifically general topology and metric topology, a compact space is a mathematical space in which any infinite sequence of points sampled from the space must eventually get arbitrarily close to some point of the space." [0, 1] is a compact space. (0.2, 0.8, 0.2, 0.8, ....) is an infinite sequence. It does not eventually get arbitrarily close to any point in [0, 1]. Surely this is not a definition of compact? Veltas (talk) 19:03, 4 June 2013 (UTC)

- Pointing out what is hopefully obvious: 0.8 is in [0,1]. Sławomir Biały (talk) 20:12, 4 June 2013 (UTC)

- Either you can take "eventually arbitrarily close" to mean eventually the rest of the sequence is a certain distance, or as you have done "there is a point in the sequence that is arbitrarily close to a point in the space", which, in fact, since the sequence is sampled from the space is implied. So either it's the wrong definition, or not a definition at all. Veltas (talk) 20:53, 4 June 2013 (UTC)

- Perhaps the confusion is that compactness isn't convergence: it doesn't require every element of that sequence to converge arbitrarily close to the same point. In your sequence above, the the odd elements get arbitrarily close to 0.2 and the even elements get arbitrarily close to 0.8. There are other examples of non-convergent infinite sequences in the article that nonetheless have a subset that gets arbitrarily close to some point in the associated compact space. --Mark viking (talk) 20:50, 4 June 2013 (UTC)

- However they have stated specifically "any infinite sequence of points sampled from the space must eventually get arbitrarily close to some point of the space" when it is only true to say "any infinite sequence of points sampled from the space must have a subsequence that gets arbitrarily close to some point of the space eventually." So the 'rough' definition is wrong. It's unfortunate because they're obviously attempting to give an intuitive notion of compactness, but the definition implies something that doesn't really make sense. Veltas (talk) 20:59, 4 June 2013 (UTC)

- No, you're reading the quantifier wrong. There's a point in the space that the sequence eventually gets arbitrarily close to. Sławomir Biały (talk) 21:10, 4 June 2013 (UTC)

- What quantifier? You seem to suggest that the notion of a sequence eventually getting arbitrarily close to a point is the notion of a convergent subsequence, but intuitively to me a sequence eventually getting arbitrarily close to a point is convergence. When a property of a sequence is 'eventual' in a sequence, I take it to mean it applies to all points beyond a certain position in the sequence. Combined with 'arbitrarily close' we have the literal definition of convergence. The only way I can read this sensibly and garner a sense of the idea of compactness is as: "a compact space is a mathematical space in which any infinite sequence of points sampled from the space must have an infinite subsequence eventually get arbitrarily close to some point of the space," or perhaps better yet; "in a compact space any sequence of the space must have a subsequence that converges to a point in the space", since a sequence may as well be finite (and then it's equivalent to an infinite sequence with the end element repeated at the end), or something of the sort. Veltas (talk) 21:29, 4 June 2013 (UTC)

- While it's not wrong, you have a point--it could be made more clear. I have edited the sentence to read: "In mathematics, specifically general topology and metric topology, a compact space is a mathematical space in which any infinite sequence of points sampled from the space
*has a convergent subsequence*that must eventually get arbitrarily close to some point of the space." This is consistent with the definition given in Sequentially compact space an in the online sources I've skimmed. Thanks, --Mark viking (talk) 21:21, 4 June 2013 (UTC)

- While it's not wrong, you have a point--it could be made more clear. I have edited the sentence to read: "In mathematics, specifically general topology and metric topology, a compact space is a mathematical space in which any infinite sequence of points sampled from the space

- I agree with User:Veltas that the sentence is really ambiguous. What about something like
- "must eventually come arbitrarily close, infinitely many times, to some point of the space"
- "must eventually come arbitrarily close, at some times, to some point of the space"

- or anything of the sort, in a better English than mine.
- Bdmy (talk) 21:37, 4 June 2013 (UTC) Bdmy (talk) 21:44, 4 June 2013 (UTC)

- I agree with User:Veltas that the sentence is really ambiguous. What about something like

I've just realised the whole first line is wrong, it's the definition of sequential compactness. Unless it's made clear that this is a 'watered-down' version for an intuitive glance at its meaning it should be exchanged for the actual definition. That is, for all open covers there is a finite subcover, otherwise the first line is in fact wrong regardless of our different understandings. Sequential compactness is equivalent for metric spaces, at least, but it's crucial that the difference is recognised. Veltas (talk) 21:54, 4 June 2013 (UTC)

- I've changed it so that we don't give the wrong meaning anymore, and at least it's clear that there is a difference at least, and maybe the example helps give a 'feel' for what compactness is. It's clear (as has been noted by many others in this talk page) that there is still much to do to clean up the start of this article. Veltas (talk) 22:34, 4 June 2013 (UTC)
- The first paragraph of the lead is quite clear that there are different notions of compactness, which are only equivalent in good cases. Sławomir Biały (talk) 15:00, 5 June 2013 (UTC)

## Changing the lead completely?

The lead of the article has multiple issues; some of them have not been quoted in the lengthy preceding discussions. The lead is too long. It too technical: to understand the first sentences, one needs to well know what is a cover and understand what means "must eventually, infinitely often, get arbitrarily close". It give an undue weight to one definition of compactness. It is confusing by using a language which is wrongly supposed to be more accessible to the layman. For example "mathematical space": somebody who knows what is a topological space will ask "what is this strange object", and people who do not know anything on topological spaces do not understand also this term (as well as the remainder of the lead). The main issue is probably that nothing is said about why compactness means in practice, why it is important and why it is useful (the context). IMHO, to solve all these issue the lead has to be completely rewritten. I suggest something like the following:

- In mathematics, a
**compact space**or**compact set**is a topological space satisfying an important property, the**compactness**, which is used in many branches of mathematics. Roughly speaking, a compact space is a space where every limit that may be hoped for exists in the space. This means that there is no "lacking point" (the set of the rational numbers is not compact as is a limit of rational numbers and is not rational) and there is no "endless path" in the space (the number line is not compact).

- The compact subspaces of
**R**^{n}are the bounded closed subsets. For more general cases, compactness may be defined intrinsically, that is without referring to a larger space in which the compact space is embedded. There are several definitions, which need some further, commonly satisfied, conditions to be satisfied.

- The importance of compactness lies in the number of useful properties that may be deduced from it. For example, every continuous function from a compact space into the real numbers reaches its maximum and its minimum; this means that there exist two points in the compact set, where the value of the function is, respectively, maximum and minimum.

This is only a first draft, but I believe that it solves most of the issues of the present lead. D.Lazard (talk) 10:55, 31 October 2013 (UTC)

- P.S. The proposition in #How about starting with closed and bounded in R^n? has been posted when I was writing mine. They are thus completely independent. D.Lazard (talk) 11:05, 31 October 2013 (UTC)

- I have some issues with an attempt to completely rewrite the lead. First, the current version of the lead is intended to be accessible to laymen. Indeed, many non-mathematicians have commented here that the current version of the lead is one of the most understandable introductions to the topic that they were able to find. That is clearly an important data point. Most of the negative comments come from students who are upset that the definition described in the lead does not agree exactly with the definition given in their topology textbook. But our aim must be to write "one level down". Compactness is usually introduced in an upper level undergraduate course, so the lead should be accessible to a student of calculus. Second, the current version of the lead summarizes the article, including a historical perspective on the concept and summary of the article. Finally, the lead as currently written was the product of many discussions here. I think it would be inappropriate to disregard these efforts entirely. Sławomir Biały (talk) 11:21, 31 October 2013 (UTC)

- The virtues of the lead, as it stands, are those Slawomir Bialy mentions. It is also all these virtues that makes it way too long. After the lead comes a lengthy introduction (essentially an extension of the lead), and
*then*we get to the definition of compactness. YohanN7 (talk) 15:08, 8 November 2013 (UTC)- I feel like the lead is as long as it needs to be to provide an
*accessible*overview that defines the topic of the article and summarizes it. Some trimming might be possible, but it is three paragraphs long per the recommendations of WP:LEAD. However, I would agree that with the present form of the lead, the introduction section is almost entirely redundant. It might be better to replace this with a section including more examples of compact spaces. Charles Matthews makes an interesting suggestion that would allow for the inclusion of images as well in such a section. I wouldn't object if someone wants to start work on such an enterprise. Sławomir Biały (talk) 15:22, 8 November 2013 (UTC)

- I feel like the lead is as long as it needs to be to provide an

- The virtues of the lead, as it stands, are those Slawomir Bialy mentions. It is also all these virtues that makes it way too long. After the lead comes a lengthy introduction (essentially an extension of the lead), and

## Gloss for "bounded" (cube/ball/sphere)

So while I reverted this edit, it did kind of raise a good point. What's the best way to gloss the word "bounded", assuming we want "closed and bounded" to have a prominent place in the lead?

I put it as "contained within some finite cube of the appropriate dimension"; the IP editor changed "cube" to "sphere". The problem with that is that a sphere, technically, is just the *surface* of a ball. So it would be correct to put "ball" or "spherical ball", but that would itself need explanation, because not everyone knows what a "ball" is. And if you have to explain your explanations too many times, it gets tiresome to read.

But it can't be denied that the "ball" definition is probably the more common one in the literature.

Suggestions? --Trovatore (talk) 01:06, 13 November 2013 (UTC)

- The phrase "inside a sphere" means the interior of the ball, so I don't see a problem with that expression. My reason for not changing it originally was that I wanted to avoid "ball", and I anticipated some objection to the word "sphere". I would like to find an east enough wording that avoids "of the appropriate dimension", regardless of the choice here. "Inside a hypersphere" possibly? Sławomir Biały (talk) 16:20, 13 November 2013 (UTC)
- I suppose you
*could*read the word "inside" in that way, but I submit that it's confusing. It's a natural reading in English but not a very natural one in mathematical English — the more "mathematically natural" one would be "is a subset of the sphere", which is not what we want. So I prefer to avoid "sphere". --Trovatore (talk) 19:27, 13 November 2013 (UTC)- I'd personally read "interior of the ball" as the
*open*ball. Probably best to avoid such a phrase. YohanN7 (talk) 20:41, 13 November 2013 (UTC) - I would read "inside a sphere" as "within a fixed distance to the origin". But perhaps this is what the article should say? Sławomir Biały (talk) 11:58, 14 November 2013 (UTC)
- What origin? We've mentioned Euclidean space; we haven't said it has a distinguished origin. And I still think "inside" is problematic. But I'm starting to think "spherical ball" is not too bad — correct for those who understand the distinction, reasonably intuitive to those who don't. --Trovatore (talk) 23:14, 14 November 2013 (UTC)
- Using "spherical ball" seems good to me. Paul August ☎ 00:28, 15 November 2013 (UTC)
- I've tried out an alternative wording that seems simpler than all of the options presented thus far since it avoids the awkwardness of "the appropriate dimension". Sławomir Biały (talk) 00:45, 15 November 2013 (UTC)

- Using "spherical ball" seems good to me. Paul August ☎ 00:28, 15 November 2013 (UTC)

- What origin? We've mentioned Euclidean space; we haven't said it has a distinguished origin. And I still think "inside" is problematic. But I'm starting to think "spherical ball" is not too bad — correct for those who understand the distinction, reasonably intuitive to those who don't. --Trovatore (talk) 23:14, 14 November 2013 (UTC)

- I'd personally read "interior of the ball" as the

- I suppose you