## Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

## Site Tag Cloud

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

• CommentRowNumber1.
• CommentAuthorTodd_Trimble
• CommentTimeOct 16th 2010

As an outcome of recent discussion at Math Overflow here, Mike Shulman suggested some nLab pages where comparisons of different definitions of compactness are rigorously established. I have created one such page: compactness and stable closure. (The importance and significance of the stable closure condition should be brought out better.)

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeJul 18th 2014

I have added to the Properties-section at compact space that in Hausdorff space every compact subset is closed.

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeMar 31st 2016
• (edited Mar 31st 2016)

I have tried to edit compact space just a little for readability.

For one, I tried to highlight nets over proper filters just a tad more (moving the statements for nets out of parenthesis, adding reminder to the equivalence via their eventuality filters), just so that the entry gets at least a little closer to giving the reader the expected statement about convergence of sequences.

Also I pre-fixed the text of all of the many equivalent definitions by “$X$ is compact if…” to make it read more like a text meant for public consumption, and less like a personal note.

• CommentRowNumber4.
• CommentAuthorTodd_Trimble
• CommentTimeMar 31st 2016

Looks good!

• CommentRowNumber5.
• CommentAuthorMike Shulman
• CommentTimeNov 19th 2016

I am confused about the constructive notion of “compact locale”. In the Elephant just before C3.2.8, Johnstone says

If $X$ is a locale in a Boolean topos, then the unique locale map $\gamma:X\to 1$ is proper iff $\gamma_*$ preserves directed joins… Constructively, the Frobenius reciprocity condition [$f_*(U\cup f^*(V)) = f_*(U)\cup V$]… is a nontrivial restriction even on locale maps with codomain 1; so we include it in the definition of compactness — that is, we define $X$ to be compact if $X\to 1$ is proper.

However, as far as I can tell this additional Frobenius condition is not included in the general definition of proper geometric morphism, which just says (in the stack semantics of $E$) that $f:F\to E$ preserves directed unions of subterminals. It makes sense that some extra condition is necessary when speaking externally about a general locale map $f:X\to Y$, since “$f_*$ preserves directed unions” is not internalized to the stack semantics of $Sh(Y)$. However, when talking about a map $X\to 1$, we should already be in the stack semantics of $Sh(1) = Set$. So I don’t understand why we need the Frobenius condition separately in the constructive definition of “compact locale”. Johnstone doesn’t give an example, and I don’t have Moerdijk or Vermuelen’s papers.

On the other hand, I don’t see how to prove the Frobenius condition either. I have looked through section C3.2 of the Elephant as carefully as I can, and I haven’t been able to extract an actual proof that a proper geometric morphism between localic toposes satisfies the Frobenius condition to be a proper map of locales. Prior to defining proper geometric morphisms, he says

…we may reasonably define a geometric morphism $f:F\to E$ to be proper if… $f_*(\Omega_F)$ is a compact internal frame in $E$. What does it mean to say that an internal frame… is compact in a general topos $E$? For the case $E=Set$, we saw how to answer this in topos-theoretic terms in 1.5.5: it means that the direct image functor $Sh(X) \to Set$… preserves directed colimits of subterminal objects.

and then proceeds to define $f:F\to E$ to be proper if “$f_*$ preserves directed colimits of subterminal objects” is true in the stack semantics of $E$. But C1.5.5 used “$f_*$ preserves directed unions” as a definition of “compact locale”, which the first quote above claims is not sufficient constructively, i.e. over a general base $E$. So I am confused; can anyone help?

• CommentRowNumber6.
• CommentAuthorMike Shulman
• CommentTimeNov 30th 2016

Ok, I found Vermeulen’s paper, and I believe he has a constructive proof that if $r:X\to 1$ preserves directed joins then it satisfies the Frobenius condition. Suppose $U\in O(X)$ and $V\in O(1) = \Omega$; we must show that if $r_\ast(U \cup r_\ast(V))$ is true, then so is $r_\ast(U) \cup V$. Note that $r_\ast(W)$ is the truth value of the statement “$W=X$”, while $r^*(P) = \bigcup \{ X \mid P \}$. Suppose $r_\ast(U \cup r_\ast(V))$, i.e. that $U\cup \bigcup \{ X \mid V \} = X$. Now consider the set $\{ W\in O(X) \mid V \vee (W\le U) \}$; this is evidently directed, and our supposition in the last sentence says exactly that its union is $X$. Therefore, if $X$ is compact in the sense that its top element is inaccessible by directed joins, there exists a $W$ such that $V \vee (W\le U)$ and $X\le W$. In other words, either $V$ or $X\le U$, i.e. either $V$ or $r_\ast(U)$, which is what we wanted.

So my current conclusion is that the first Elephant quote above is wrong about the Frobenius condition being an additional restriction constructively. This did seem like the most likely resolution, since otherwise the definition of proper geometric morphism would probably be wrong, which seems unlikely.

• CommentRowNumber7.
• CommentAuthorMike Shulman
• CommentTimeDec 1st 2016

I have added this proof to covert space, with pointers to it from compact space and proper map.

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeApr 23rd 2017
• (edited Apr 23rd 2017)

There seems to be something wrong right at the beginning of compact space:

After def. 2.1, the usual definition about existence of finite open subcovers, there is def. 2.2 which is the immediate reformulation in terms of closed subsets: a collection of closed subsets whose intersection is empty has a finite subcollection whose intersection is still empty.

But this def. 2.2 is introduced with the remark that it needs excluded middle to be equivalent to 2.1, which is not true.

Probably what that remark about excluded middle was meant to refer to is instead the further formulation in terms of closed subsets, the one which says that a collection of closed subsets with the finite intersection property has non-empty intersection.

[edit: I have added what seems to be missing at compact space to finite intersection property]

• CommentRowNumber9.
• CommentAuthorTodd_Trimble
• CommentTimeApr 23rd 2017
• (edited Apr 23rd 2017)

How are closed sets being defined here?

If we are defining closed sets to be precisely the complements of open sets (in symbols, $\neg U$), then I can see that the usual open set formulation implies the closed set formulation you just mentioned: if $\bigcup_{i \in I} U_i = X$, then surely $\bigcap_{i \in I} \neg U_i = \neg \left(\bigcup_{i \in } U_i \right) = \neg X = \emptyset$ since $\neg$ takes unions to intersections.

But then how would we turn this implication around (to assert equivalence of the two formulations)? Without excluded middle, I don’t see how every open set $U$ would be the complement of a closed set.

• CommentRowNumber10.
• CommentAuthorUrs
• CommentTimeApr 23rd 2017

Oh, okay. I’ll make all that explicit in the entry now.

• CommentRowNumber11.
• CommentAuthorUrs
• CommentTimeApr 23rd 2017

Okay, I have edited statement and proof at compact+space#fip. I made explicit the use of excluded middle for identifying opens with complements of closed subsets, and a second use of excluded middle for getting what is usually labeled “fip”, which is the contrapositive of what was formerly stated here.

• CommentRowNumber12.
• CommentAuthorMike Shulman
• CommentTimeApr 23rd 2017

FWIW, constructively there are at least two distinct definitions of closed set.

• CommentRowNumber13.
• CommentAuthorUrs
• CommentTimeMay 14th 2017

I have added some elementary details at compact space – Examples – General, such as the proof that closed intervals are compact.

• CommentRowNumber14.
• CommentAuthorUrs
• CommentTimeMay 15th 2017

I have added the statement about unions and intersections of compact subspaces, here

• CommentRowNumber15.
• CommentAuthorTodd_Trimble
• CommentTimeJun 3rd 2017

In accordance with a recent discussion, I moved the detailed elementary proof of the example of closed intervals to its own page. Also, I added the example of cofinite topology to compact space.

• CommentRowNumber16.
• CommentAuthortphyahoo
• CommentTimeJul 2nd 2017
Proposition 3.2. (complements of compact with open subspaces is compact)

seems wrong.

Take [0,3] as compact space and (1,3) as subspace of R1.

The complement is two half open spaces: [0,1) and (2,3] and by nonexample 4.6 "half open intervals are not compact"

Either proposition 3.2 is wrong or I am missing some intuition. A proof or a reference to a proof would help clear this up if I am wrong here.
• CommentRowNumber17.
• CommentAuthortphyahoo
• CommentTimeJul 2nd 2017
• (edited Jul 2nd 2017)
Definition 2.4, point 4 is
"For every topological space (Y,τY) (Y,\tau_Y) the projection map out of the product topological space πY:(X×Y,τX×Y)→(Y,τY) \pi_Y \;\colon\; (X \times Y, \tau_{X \times Y}) \to (Y, \tau_Y) is a closed map.

and proof of this is stated as
"The proof of the equivalence of statement 4 is discussed at closed-projection characterization of compactness."

further down

Definition 2.10. (closed-projection characterization of compactness) is
"X X is compact iff for any space Y Y, the projection map X×Y→Y X \times Y \to Y out of their Cartesian product is closed (see e.g. Milne, section 17)."

Aren't these two ways of saying the same thing?

Proposed fix: 2.4, point 4 and its proof should be deleted from 2.4 and merged to 2.10.

Possible issue, assuming validity of above fix:

Does 2.10 then require excluded middle?
It is claimed that point 2.4 point 4 requires excluded middle, but it is not clear why this is.
In fact linked "closed-projection characterization of compactness" claims a means of circumventing both excluded middle. My guess is excluded middle is not required but I don't know.
• CommentRowNumber18.
• CommentAuthorMike Shulman
• CommentTimeJul 2nd 2017
• (edited Jul 2nd 2017)

Re: #16, $[0,3] \setminus (1,3)$ is $[0,1] \cup \{3\}$, which is compact.

• CommentRowNumber19.
• CommentAuthorMike Shulman
• CommentTimeJul 2nd 2017

Re: #17, I think you’re right that 2.4(4) and 2.10 are redundant. I don’t have an opinion on which of them should be kept.

• CommentRowNumber20.
• CommentAuthorUrs
• CommentTimeJul 3rd 2017

I have now spelled out a detailed proof of prop. 3.2 here.

• CommentRowNumber21.
• CommentAuthorUrs
• CommentTimeJul 3rd 2017

Regarding the duplication of the statement of the closed-projection characterization of compactness:

I have removed item 4 from prop. 2.4, but I also moved the former “Definition” 2.10 up to what is now prop. 2.5, so that it is still the next statement after item 3 of prop. 2.4.

(Todd should please have a look.)

My main request about this entry is: Somebody should turn the long list of “Definitions” 2.7 to 2.13 of compactness into a list of propositions that state that certain statements are equivalent.

• CommentRowNumber22.
• CommentAuthorTodd_Trimble
• CommentTimeJul 3rd 2017

Thanks, Urs. It looks good to me. Does the mention of excluded middle now make sense to tphyahoo? To circumvent excluded middle (as in closed-projection characterization of compactness), one has to change the statement to say the dual image operator $\forall_\pi: P(X \times Y) \to P(Y)$ takes open sets to open sets; the statement as it is, that the direct image along projection takes closed sets to closed sets, is equivalent to the other statement by De Morgan duality, but that’s where excluded middle comes in.

• CommentRowNumber23.
• CommentAuthortphyahoo
• CommentTimeJul 3rd 2017
Thank you yes, much clearer now.
• CommentRowNumber24.
• CommentAuthortphyahoo
• CommentTimeJul 3rd 2017
After a bit of grammar cleanup, Definition 2.12 now seems more obviously incoherent.

First, we need a definition of stably closed. The colon suggests a definition, but I believe it may be an error. If it is not an error, this could be clarified by defining it on its own concept page (currently nonexistent).

If it is an error as I suspect and "stably closed" is unrelated to the closed projection characterization, then:

1) "stably closed" should be its own definition and perhaps its concept page should be created as well.

2) Then definition 2.12 and 2.13 should be merged or moved up to be closer to Proposition 2.5 so that all the stuff about closed projection is in one place.
• CommentRowNumber25.
• CommentAuthortphyahoo
• CommentTimeJul 3rd 2017
I think the link to the "logic" page in

"a logical characterisation of compactness is used in Abstract Stone Duality:"

should be removed.
• CommentRowNumber26.
• CommentAuthortphyahoo
• CommentTimeJul 3rd 2017
It is not clear if the assumption of the ultrafilter theorem applies just to 2.7, or to 2.8 and 2.9 as well or even 2.10 and 2.11.

I believe that Section 2.10 and 2.11 is unrelated to filters/nets but this should be made more obvious with better section numbering and breaks.

I also believe section 2.12 is unrelated to the earlier section, but I noted this before.
• CommentRowNumber27.
• CommentAuthortphyahoo
• CommentTimeJul 3rd 2017
I think section 2.7 through 2.9 should be grouped under

Proposition N.N. (compactness in terms of filters)

If these are all equivalent (and ultrafilter theorem requirement applies to all) then the proposition should be along the lines of "the following three statements are equivalent."

Similarly, section 2.10 and 2.11 should be grouped under

Propsition N.N (compactness in terms of open sets)

Possibly Definition 2.12 and 2.13 then get merged under proposition 2.5.

Possibly there is also a new proposition about proper maps, that is broken out of 2.12.
• CommentRowNumber28.
• CommentAuthorUrs
• CommentTimeJul 3rd 2017

I agree, with #24, that’s what I was referring to at the end of #21.

This needs to go to the attention of Toby and Todd, I think. The request would be: Turn these terse remarks into something a little more self-contained and inviting.

• CommentRowNumber29.
• CommentAuthortphyahoo
• CommentTimeJul 3rd 2017
I am having trouble making the above changes because of struggles with wiki markup, however, to be more explicit I think we should have something like

Proposition 2.6. (compactness in terms of filter or net convergence)

Let X be a topological space. Assuming the ultrafilter theorem (a weak form of the axiom of choice), then the following are equivalent:

1 X is compact in the sense of def. 2.2.
2-.... reformulation of definitions 2.6-2.9
• CommentRowNumber30.
• CommentAuthortphyahoo
• CommentTimeJul 3rd 2017
then

Proposition 2.7 (compactness in terms of frames of opens)

Let X be a topological space. Then the following are equivalent:
1 X is compact in the sense of def. 2.2.
2-.... reformulation of definitions 2.10 and 2.11
• CommentRowNumber31.
• CommentAuthortphyahoo
• CommentTimeJul 3rd 2017
Proposition 2.8 same thing for stably closed, if this is a stand alone concept unrelated to closed projection characterization of compactness.
• CommentRowNumber32.
• CommentAuthortphyahoo
• CommentTimeJul 3rd 2017
Proposition 2.5 can then perhaps be reforumulated as

Let X be a topological space. Then the following are equivalent:

1 X is compact in the sense of def. 2.2
2 current contents of proposition 2.5
3-... reformulation of definitions 2.12 and 2.13
• CommentRowNumber33.
• CommentAuthorTodd_Trimble
• CommentTimeJul 3rd 2017

Okay, I’ll begin having a look. Much of this was written (I think) by Toby quite a while ago, and his presence here has lately become more sporadic.

Please be patient though. I am finding some of the tone harsh (e.g., “now seems more obviously incoherent” – that I find excessive), and it’s a bit of a barrage of comments now to process.

• CommentRowNumber34.
• CommentAuthortphyahoo
• CommentTimeJul 3rd 2017
I'm very sorry about my tone.

I am extremely enthusiastic about constructive math (an amateur obviously, I admit). And I admire what the nlab has accomplished collectively, as well as the individual contributors here and on the cluster of pages where I lurk. Todd / Urs / Toby: thanks for everything you've done.

The reason I sent so many comments is I wanted to lay a breadcrumb trail for future edits to the wiki, since I was unable to edit the wiki myself to my satisfaction. It keeps cutting off the page halfway due to some formatting glitch I can't track down. And there doesn't seem to be a preview functionality.

If there is any advice or links about editing / maintaining the wiki I will review and try to do better in the future.

BTW, I am https://www.linkedin.com/in/thomashartman1/ just to put a face to the edits. I have been interested in constructive math for some years as a software developer doing haskell / coq for personal interest and some for work. Lurker on nlab for years I guess. I am primarily interested in constructive characterization of the reals, probably following the abstract stone duality path.
• CommentRowNumber35.
• CommentAuthorTodd_Trimble
• CommentTimeJul 3rd 2017

Okay, I’ve gone through and reorganized section 2 of compact space according to comments/suggestions above and my own personal knowledge. Roughly I classified the various “definitions” (now called propositions) under three headings: elementary reformulations, via convergence, and via stability properties. There is still plenty left to do: plenty of proofs which can be filled in or farmed out to other parts of the nLab, as appropriate, and still links left to be made, among other things.

It is good that you (tphyahoo) brought your concerns to our attention, so thanks for that. I do think the article has a better shape now. The “obviously incoherent” former 2.12 is, I hope you will now see, coherent after all. I also changed the link from logic to quantification which is more precise I think.

• CommentRowNumber36.
• CommentAuthorUrs
• CommentTimeJul 4th 2017

Thanks, Todd!! That’s great.

• CommentRowNumber37.
• CommentAuthorTodd_Trimble
• CommentTimeJul 4th 2017

Something seems wrong with Proposition 3.3, that the category of compact spaces has limits. (Of course compact Hausdorff spaces have limits.) The problem is that the equalizer of two maps need not be a closed subspace; that kind of thing is true if we are working with Hausdorff spaces, but not for more general spaces.

An explicit example is where we take two maps $[0, 1] \to \{0, 1\}$ where the codomain is given the indiscrete topology, where one of the maps $f$ has $f^{-1}(1) = [0, 1/2)$, and the other is the constant map at $1$. If an equalizer in $Comp$ existed, then $\hom(1, -): Comp \to Set$ would have to preserve it, so set-theoretically it would have to be $[0, 1/2)$. The topology on $[0, 1/2)$ would have to be the same as or finer than the subspace topology in order for the equalizer map to be continuous. But if the subspace topology isn’t compact, then no finer topology would make it compact either. (Here I’m taking the contrapositive of the proposition that if $(X, \tau)$ is compact and $\tau' \subseteq \tau$ is a coarser topology, then $(X, \tau')$ is also compact.)

I guess I could go in and change it to a true statement, but I’d want to know first about the situation for compact locales. Again, for compact regular locales, there’s no problem.

• CommentRowNumber38.
• CommentAuthorMike Shulman
• CommentTimeJul 4th 2017

I feel like if compact locales had limits we would know about it. Could you do something similar to your counterexample with the Sierpinski space instead of the indiscrete 2-point space?

• CommentRowNumber39.
• CommentAuthorTodd_Trimble
• CommentTimeJul 4th 2017

Yes, it seems the same counterexample works with Sierpinski space, since $1$ is usually taken to be the open point. (Maybe this is discussed somewhere in Stone Spaces? I don’t have my copy within easy view.) So yeah, if I think about it a little longer, maybe it will become obvious that compact frames lack coequalizers, or maybe you already see that’s true.

• CommentRowNumber40.
• CommentAuthortphyahoo
• CommentTimeJul 8th 2017
From the introduction a compact space

"is a kind of ultimate topological expression of the general idea of a space being “closed and bounded”"

but is it true that all compact spaces are closed with regard to some topology, or only metric spaces, or only euclidean spaces?

Proposition 3.1 (2) has the precondition that "if the compact spaces are also closed" then the claim about inersections.

Well, if all compact spaces are closed then this precondition can be dropped as the claim applies to all compact spaces.

But if not all compact spaces are closed, perhaps this should be dropped from the idea section, or qualified "In metric spaces..."
• CommentRowNumber41.
• CommentAuthorTodd_Trimble
• CommentTimeJul 8th 2017

That “Idea” section should be read at a very intuitive level; the language is somewhat fuzzy (I suppose by design). The author of those words should be granted some poetic license.

Any space, compact or not, is closed in itself, so under strict interpretations some of the language of the Idea section won’t make a lot of sense. The real sense of that passage is concentrated in the words “every net has an accumulation point” (that certainly has a precise meaning), and that should be the main takeaway from what the author is trying to convey. The rest of it seems to be a simple appeal to the archetypal image we all carry around in our heads of compact sets: sets in $\mathbb{R}^n$ which are closed and bounded, and the author is trying to draw a connection between that intuitive conception of “closed and bounded”, based on that (finite-dimensional) picture, and the more precise mathematical conception “every net must have an accumulation point”.

In infinite-dimensional Hilbert space (for example), “closed and bounded” do not imply compact. So that intuition comes with a (largish) grain of salt! In fact the word “bounded” isn’t quite a topological concept; it makes sense for metric spaces but it doesn’t have a meaning for general topological spaces.

Turning to Proposition 3.1: now we’re doing mathematics, not waving intuitive wands. As it happens, compact subspaces of Hausdorff spaces are closed (and Hausdorff spaces make up the majority of spaces one encounters when first learning topology), but in general compact subsets need not be closed in the ambient space. Thus the closure hypothesis has to be inserted by hand for the proposition to work in the generality given there.

Anyway, you may be right that the Idea section is (for some readers anyway) more confusing than enlightening. It’s hard to say, but perhaps the opening should be reconsidered.

• CommentRowNumber42.
• CommentAuthortphyahoo
• CommentTimeJul 8th 2017
"by boundedness it cannot escape, and by closure the point is in the space."

can "it" here be replaced by "accumulation point"? I think that would be clearer.
• CommentRowNumber43.
• CommentAuthorTodd_Trimble
• CommentTimeJul 8th 2017

No, the “it” here is the net (nets are generalized sequences).

• CommentRowNumber44.
• CommentAuthortphyahoo
• CommentTimeJul 8th 2017

"In typical spatial domais (such as R^n) compactness is a kind of ultimate topological expression of the general idea of a space being “closed and bounded”: .........
• CommentRowNumber45.
• CommentAuthortphyahoo
• CommentTimeJul 9th 2017
What do you think of singly linking "closed and bounded" to the heine borel theorem section (4.5) rather than linking "closed" and "bounded" separately to the two concept pages as is currently the case.

Doesn't this give a better intuition?
• CommentRowNumber46.
• CommentAuthortphyahoo
• CommentTimeJul 9th 2017
Also it seems like many of the "Examples" (such as heine borel) fix in better in the Definitions / Characterizations section.
• CommentRowNumber47.
• CommentAuthorTodd_Trimble
• CommentTimeJul 9th 2017
• (edited Jul 9th 2017)

Re #46: no, the definitions/characterizations section is general. Heine-Borel is for a specialized set of circumstances and belongs to Examples.

• CommentRowNumber48.
• CommentAuthorTodd_Trimble
• CommentTimeJul 9th 2017

Okay, I tweaked the opening paragraph just a little. I don’t agree with joining “closed” to “bounded” as in #45, but I did mention Heine-Borel. Hopefully it’s now clear that it’s more about nets and convergence than it is about being closed and bounded.

• CommentRowNumber49.
• CommentAuthortphyahoo
• CommentTimeJul 9th 2017
Much clearer.

The verb "captures" is used twice, which makes it a little hard to concentrate on exactly what is being captured. I reworded to avoid this.
• CommentRowNumber50.
• CommentAuthorTodd_Trimble
• CommentTimeJul 9th 2017

But there was no mention of compactness in the 2nd paragraph of your edit; attention was momentarily deflected away to talk about “closed and bounded”. So I had to reword again.

Please do not remove the parentheses. They signal that a side remark is being made, which the reader can pursue if she likes, but the main focus needs to be kept on what the property is about.

Thank you for your input, but now that some clarity has been reached, I suggest that we not spend a lot of time on what Feynman called “wordsmithing”. I’ll add that I made a change suggested by comment #37.

• CommentRowNumber51.
• CommentAuthorUrs
• CommentTimeJul 9th 2017

I took a look and ended up making some last changes myself to the Idea section:

• made explicit that “everything” in the first sentence referred to sequences and nets

• linked the line about not needing and ambient space for the definition with the line saying that nevertheless one does often consider compact subspaces;

• after the claim that one likes to consider compact Hausdorff spaces, I added one reason

• grouped the two lines about compact locales together, now at the end of the Idea-section.

• CommentRowNumber52.
• CommentAuthorTodd_Trimble
• CommentTimeJul 9th 2017

Those are useful improvements, Urs – thanks.

• CommentRowNumber53.
• CommentAuthorTodd_Trimble
• CommentTimeJul 9th 2017

Urs, in the idea section, did you really mean to say “paracompact” and not “locally compact”?

• CommentRowNumber54.
• CommentAuthorUrs
• CommentTimeJul 9th 2017

Argh. :-)

Thanks. I have fixed it, and also added pointer to the proof.