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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.)
I have added to the Properties-section at compact space that in Hausdorff space every compact subset is closed.
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.
Looks good!
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?
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.
I have added this proof to covert space, with pointers to it from compact space and proper map.
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]
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.
Oh, okay. I’ll make all that explicit in the entry now.
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.
FWIW, constructively there are at least two distinct definitions of closed set.
I have added some elementary details at compact space – Examples – General, such as the proof that closed intervals are compact.
I have added the statement about unions and intersections of compact subspaces, here
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