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at sober space the only class of examples mentioned are Hausdorff spaces. What’s a good class of non-Hausdorff sober spaces to add to the list?
Affine schemes with the Zariski topology.
Edit: in fact any scheme!
Thanks. I’ll add that to the entry. What’s a good citation for this?
Not an original source, but http://stacks.math.columbia.edu/tag/01IS
Thanks! I have added that to schemes are sober and cross-linked
I added to schemes are sober the statement that by only considering maximal ideals in the definition of spectrum of a ring, one doesn’t get a sober space.
At sober topological space I made the definition of “topological space with enough points” a numbered environment (here) so such as to be able to point to it from further below in the text where it is being invoked.
I added the statement that the three conditions listed are meant to be equivalent. I hope that’s right.
David, and I’ll add the statement that the sobrification of the “maximal spectrum” is the usual “prime spectrum”.
briefly added first counter-examples (here) showing that the classes of $T_1$-spaces and sober spaces do not contain each other.
I have added to sober topological space statement and direct proof that for sober topological spaces $(X,\tau_X)$, $(Y, \tau_Y)$ then frame homomorphisms $\tau_X \leftarrow \tau_Y$ are inverse images of continuous functions – here.
I was thinking of giving this statement and proof its own page “sober spaces are the locales with enough point”, because this statement is mentioned in several $n$Lab entries, and always without proof. But I haven’t created that stand-alone page yet.
You don't have to bother with preserving inclusions in a frame homomorphism; that follows from preserving binary intersections and/or unions. And from the perspective of frames as algebras, it's an unnatural condition anyway. (From the perspective of frames as posets it's quite natural but still redundant.)
You've put a nonexample under Examples. (In fact, there was already a nonexample in the desired class —T₁ but not sober— under Nonexamples.) Maybe Examples and Nonexamples should be combined?
Maybe Examples and Nonexamples should be combined?
Thanks, yes. I have merged them.
You don't have to bother with preserving inclusions in a frame homomorphism; that follows from preserving binary intersections and/or unions.
True, thanks. I have split off preservation of inclusions as a remark, here.
[wrong thread, deleted]
I have decomposed the proof that continuous maps between sober spaces are equivalent to the corresponding frame homomorphisms by splitting off the lemma – here – that irreducible closed subsets are equivalently the frame homomorphisms to the opens of the point.
That’s in order to re-use this lemma in the construction and proof of the soberification reflection further below in the entry.
I started to write out the details for the soberification reflection on topological spaces. (here) Needed to interrupt for a bit.
I have now finished typing out full details of the soberification reflection, here.
Maybe it needs some more polishing, will look into it tomorrow.
It’s great to have these details on the lab. For what it’s worth, though, I think it’s a shame to state and prove theorems like this so non-constructively, when a large part of the point of locales is to be constructive.
I agree with Mike, but perhaps this can wait until the course is over? At least references to constructive proofs might be good for now.
Yes, sorry, what I need for my purposes right now is a very light-weight discussion of these things, suitable for readers with no sophisticated background, who have only just been introduced to the concept of topological spaces as such.
In particular I want to talk about sober topological spaces without actually introducing locales as such.
But there is plentry of room on the nLab. We may still accompany these classical proofs with any number of variants.
suitable for readers with no sophisticated background,
Is there an element in what Mike’s saying of the thought that if you raise students a certain way, constructive proofs with locales would just appear simpler than a classical approach with topological spaces? But I guess the course is embedded in a classical syllabus.
But I guess the course is embedded in a classical syllabus.
Yes. I am afraid I will already have to explain myself for talking about sober spaces.
On the other hand, presently most discussion of sober topological spaces in the literature is absolutely buried under a heap of localic stuff, under loads of extra terminology, that is way overkill for the simple statements about sober spaces that the classical topologist may care about. I suggest to view a classical discussion of sober spaces as an excellent motivation for, later, considering locales.
In particular I think it should be useful to classically motivate sober spaces, in turn, not via locales, as mostly done, but by the simple observation that $T_0$ is equivalent to the function $Cl(\{-\}) \colon X \to IrrClSub(X)$ being mono.
From this it is inevitable that one should also consider the stronger condition that $Cl(\{-\})$ is a bijection, and that’s soberness.
Then, as an afterthought, one realizes that these sober spaces are special in that their continuous functions my be characterized without reference to the underlying sets. And so one discovers a completely hands-on reason for studying locales, indepndent of foundational considerations, which are alien to most people.
That’s all quite reasonable. But such observations can equally well be made in a constructive way.
I’d be interested in seeing it done. Alas, I don’t have the leisure to look into it myself for the time being.
Much as I like doing things in a constructive way, I think that it is less obscure, particularly to beginning mathematicians, to refer to $Cl(\{-\}) \colon X \to IrrClSub(X)$. The question here is characterizing the closures of points as the irreducibly closed subsets. But that's not constructively correct.
The constructive version is $\mathcal{O}_X \cap \mathcal{N}(-)\colon\; X \;\to\; CompPrFil(\mathcal{O}_X)$, characterizing the collections of open neighbourhoods of points as the completely prime filters of open subsets. And that's a higher-order concept. We're talking about elements of $\mathcal{P}\mathcal{P}X$ instead of elements of $\mathcal{P}X$.
Maybe you’d have the energy to add this to the list of alternative characterizations at irreducible closed subset?
It occurs to me that even constructively it is still true that the closures of points are irreducibly closed. ($Cl\{x\}$ is inhabited, since $x$ is in it; and if $Cl\{x\}$ is (contained in)^{1} a union $F_1 \cup F_2$ of closed sets, then $x$ is in one of the $F_i$, hence $Cl\{x\}$ is contained in that $F_i$.) So we can still consider topological spaces such that the function $Cl\{-\}\colon\; X \;\to\; IrrClSub(X)$ is a bijection. What are they like?
Constructively, we can't assume that $F_1 \cup F_2$ is closed, so requiring $Cl\{x\} = F_1 \cup F_2$ might seem stricter than allowing $Cl\{x\} \subseteq F_1 \cup F_2$. However, even in that case, $Cl\{x\} = (Cl\{x\} \cap F_1) \cup (Cl\{x\} \cap F_2)$, and $Cl\{x\} \cap F_1$ and $Cl\{x\} \cap F_2$ are closed. ↩
@Urs #28: I added a remark. But then I noticed that those statements are already listed as the first properties.
I think that it would be best to keep these indirect nonconstructive alternative characterizations in the properties, actually.
Maybe I was wrong to point you to the entry irreducible closed subset in reaction to Mike’s #19. (But please feel invited to re-organize this entry, as you see the need.)
Consider the introduction to topology as at Introduction to Topology – 1. The design criterion is not to go into the theory of locales, instead keeping it elementary and at the point-set level. The task is to see how to replace the proofs that use excluded middle (all searchable by the string “by contradiction”) by constructive proofs.
Having that string is convenient. (I notice that you sometimes also write ‘using classical logic’ at the beginning of some of your propositions, and I have changed some of these to ‘using excluded middle and dependent choice’ or the like, as appropriate; but this is in addition to having ‘proof by contradiction’ at the end of your proofs, and I have not been altering those.)
If you want to keep locale theory out of it, do you even want to mention the characterization in terms of frame homomorphisms? I mean, you can say a lot about frame homomorphisms in various places in topology, but every time that you do, you're talking about continuous maps between locales; so if you're planning to discuss frame homomorphisms, then it will only simplify the discussion to introduce locales first. Conversely, if you don't want to get into locales, then you probably don't want to get into frame homomorphisms, which are (pretty much by definition) equally complicated.
Perhaps we should keep the brief discussion of the relationship to frames and locales under Properties and remove the detailed proof of the correspondence between irreducibly closed subspaces and certain frame homomorphisms entirely. But I mean entirely from this page, not entirely from the Lab, so presumably we should make a new page for it, perhaps irreducible closed subsets are in bijective correspondence with points of the locale of open subsets?
Perhaps this conversation should be continued at irreducible closed subspace.
so if you’re planning to discuss frame homomorphisms, then it will only simplify the discussion to introduce locales first.
Of course both the concept of frame homomorphisms nor that of locales are very simple in themselves. But I feel that introducing the concept of locales means being heavy on the terminology, with all that “join” and “meet” and “top” and “bottom” business, which is just a distraction if all we want to say is “preserves unions and finite intersections of opens”. This new terminology is good for readers with the inclination to learn a new foundational point of view, but for classical readers who don’t know yet that this might be their inclination, and who just want to see what’s special about sober spaces, I find it a distraction. Instead I’d like to say is: “Look, continuous maps between sober spaces may be characterized by referring only to the system of their opens, not to the underlying points.” Maybe I shouldn’t even introduce the terminology “frame homomorphism”, but just keep saying “maps that preserves unions and finite intersections”.
The same kind of feeling applies to category theory as a whole. I find that a common mistake in first expositions involving category theory is to explicitly pause discussion of a concrete field of mathematics to introduce all the jargon of category theory for what is ultimately a bunch of trivialities. Much better, pedagogically, to mention the concepts briefly in passing, as one works through some specific field, as in “by the way, what we just described is also called the ’category of topological spaces’ ” and “by the way, this property is also called ’being the limit of a diagram of spaces’ ” and the like.
Here it’s the same with locales. I don’t want to set up their theory formally to first readers in topology, I’d rather point out some quick classical facts which are such that every attentive reader will get a glimpse of why some more foundational change of perspective might be useful and thus maybe find motivation to dig deeper into it later.
Yeah, if you don't want to say “join” and “meet” and “top” and “bottom”, when you're clearly comfortable saying “union” and “intersection”, then don't say those other words. How is that the problem?
I changed the third definition of sober, “every irreducible closed set (non-empty closed set that is not the union of any two proper closed subsets) is the closure of a point”, to instead read “unique point”. This forces the topology to be T0.
A class of non-T1 sober spaces is given by the continuous directed complete posets under the Scott topology. (If you omit “continuous”, there is Johnstone’s counter-example.)
Another natural class is the T0 injective spaces (over subspace embeddings). T0 is needed because all indiscrete spaces are injective (but the T0 reflection of any injective is again injective). This second class is subsumed by the first, because Dana Scott proved that the injective T0 spaces are precisely the continuous lattices with the Scott topology.
Then there are the stably compact spaces, which enlarge the class of compact Hausdorff spaces to the non-T1 case. In one definition of them, sobriety is included as a requisite (sober locally compact with the compact subsets closed under finite intersections (including the nullary one)), and so this is cheating. But here are many other characterizations that don’t refer to sobriety: (1) The algebras of the prime-filter monad (Wyler? Harold Simmons?), (2) the injective T0 spaces over flat embeddings (Bob Flagg and myself) (do we need to say T0? (3) the retracts of the Spectral spaces (Johnstone).
Flat embedding was characterized by Isbell as “pro-super-split”: An embedding $e:X \to Y$ is flat $\iff$ for every finite T0 space $F$ and every continuous map $f:X \to F$ there is a largest continuous map $f':Y \to F$ (in the pointwise specialization order) extending $f$ along $e$. The “pro” is because of the finiteness, and the “super” because of “largest”. His paper is called “flat = prosupersplit”, I think. This is interesting, because you start with the finite spaces to get a class of embeddings, and then use this class of embeddings to generalize the finite spaces, getting the stably compact spaces. (Different people use different terminologies for these spaces: stably compact (compendium of continuous lattices community), compact and stably locally compact, or CSLC (Johnstone in the 1980’s), stably locally compact (Johnstone in the Elephant), skew compact (Harold Simmons).)
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