Prime ideal topology (S48) revising description

[GeoffreySangston]

There seem to be some issues with the third paragraph/sentence in https://topology.pi-base.org/spaces/S000048 (the one beginning "Any open $U \subset X$ is open...").

[JoeF131]

That's interesting. The first line claims that $\{ V_{(x)} : x \in \mathbb{Z}^+\}$ is a basis of $X$ eventhough it's already the entirety of the topology. Normally one defines the Zariski topology as the topology generated by the basis $\{V_I : I \text{ prime ideal }\}$ and I think the second section tries to explain that which is in that case redunant since the first line uses a different definition for the Zariski topology.

Additionally I found that pi-base doesn't know that S48 is Spectral, which follows imediately by the equivalent definition of spectral.

[GeoffreySangston]

So maybe instead of 'has a generic point', this trait should assert quasi-sober. S48 is also currently missing Alexandrov. Edit: Oh are you saying S48 is the Zariski topology on $\mathrm{Spec}(\mathbb{Z})$ (I've never studied these concepts properly)? Then I guess it would be better to change the description of S48 and assert Spectral.

Your comment made me realize pi-base doesn't answer the following queries (though there are still a lot of spaces in pi-base not known to be Quasi-sober or ~Quasi-sober), which could be good to address in a future PR.

π-Base, Search for has a generic point + ~Quasi-sober

π-Base, Search for has a generic point + hereditarily connected + ~quasi-sober

[prabau]

@JoeF131 I am not familiar enough with the algebraic side of things, but note that pi-base's notion of Spectral space (P75) is phrased in purely topological terms as a compact sober space where the compact open sets satisfy some specific properties. So how do we already know that S48 is spectral?

The second paragraph in https://topology.pi-base.org/properties/P000075 talking about Stone duality is just background information, but can't really be relied on for justification. I think. (And furthermore, there is currently no mention of "Zariski topology" in pi-base.)

[JoeF131]

I mean P75 states "Additionally, spectra of commutative rings endowed with the Zariski topology are spectral spaces and vice versa, although this association is not one-to-one." in the second paragraph. I unfortunately only know of spectral spaces due to my limites knowledge of algebra, so I couldn't tell you why these are equivalent, but I guess it gets proven in the reference of the description of P75. I'm going to borrow that book tomorrow and I'll search for the exact theorem proving said claim.

Edit: I found the paper PRIME IDEAL STRUCTURE IN COMMUTATIVE RINGS which proves in Theorem 6 that every spectral space is the Zariski topology of a spectrum of some commutative ring, but my knowledge of this topic is to limited to understand it.

[prabau]

Right, these are difficult results. For now, we may be better off using the topological characterization.

[GeoffreySangston]

What do we think about mentioning somewhere on the page that S48 is the underlying topological space of $\mathrm{Spec}(\mathbb{Z})$ (which I just checked)?

Also, since this is going to get replaced by quasi-sober, we should either argue quasi-sober here and replace this, or just do that in a future commit and remove this file then.

[prabau]

What do we think about mentioning somewhere on the page that S48 is the underlying topological space of Spec ( Z ) (which I just checked)?

Yes.

[GeoffreySangston]

I think I've written out the simple argument (with the language of ring theory / ideals) that $\mathrm{Spec}(\mathbb{Z})$ is quasi-sober, but maybe there's a citable reference.

[JoeF131]

I just checked the book {{doi:10.1017/9781316543870}} cited in P75. The proof that Spec(A) is a spectral space is given in section 2.5. and uses purely topology (with the exception of the definition of the Zariski topology). This should be simple enough for a proof I think.

The other direction that these are equivalent is far more difficult and as such I do agree that it shouldn't be used. (And there's really no situation where one does need it in pi-base as far as I'm aware.)

[GeoffreySangston]

I have an alternative suggestion (perhaps for a future PR, but it could be relevant because it could shape what the proof of quasi-sober should say). S48 has an algebraic description because it appears in Steen-Seebach that way, and I think that's why it seems natural to assert 'Spectral'. I think (could very well be wrong though) it can also be described as a kind of 'cone' over the cofinite topology on $\omega$ (S15) which appends a generic point. I'm not sure what the standard name for this cone construction is, but after briefly looking at this page, I think it's the closed extension of S15 by a point. Maybe in order to promote the general topology style of the website, it would be better to define S48 by this, and then refer to the algebraic descriptions via an alias / secondary description?

(More broadly, the suggestion is to describe S48 more topologically if possible, even if the description I gave above is not correct.)

[prabau]

I think both of your suggestions above would be valuable improvements.

Namely, add an algebraic description of the space as $\mathrm{Spec}(\mathbb{Z})$, backed up with a good reference (wikipedia?) showing it's just an example of much more general contruction. Adding $\mathrm{Spec}(\mathbb{Z})$ as an alias would make sense to me.

And also add a more topological direct description, which is kind of mentioned in item 1 of the counterexample in S&S.

I occasionally see the words "non-Hausdorff cone". No idea what that is, but would that be related to this construction?

Edit: sample use of "non-Hausdorff cone": https://math.stackexchange.com/questions/2132332/the-poset-vs-zariski-topology-on-specr

See Definition 4.6 in https://arxiv.org/pdf/math/0702198 (originally from McCord 1966). Same as Def 1.8.2 in http://math.uchicago.edu/~may/FINITE/FINITEBOOK/FINITEBOOKCollatedDraft.pdf

It seems that's the construction you are referring to. https://en.wikipedia.org/wiki/Extension_topology mostly repeats the verbiage from S&S. Would be good if it also mentioned "non-Hausdorff cone".

[GeoffreySangston]

Namely, add an algebraic description of the space as Spec ( Z ) , backed up with a good reference (wikipedia?) showing it's just an example of much more general contruction. Adding Spec ( Z ) as an alias would make sense to me.

And also add a more topological direct description, which is kind of mentioned in item 1 of the counterexample in S&S.

I tried to do this. I may have overstepped by changing the name of the space to prioritize the non-algebraic description. I can easily revert it back and swap the orders of the descriptions if that's what we want. I thought it would be wiser to upload this commit to see what we thought first, and then update the traits to reflect the new primary definition; to that end, hopefully Item 21 of the closed extension topology section of Steen-Seebach will be helpful.

I removed the reference to Wikipedia for the closed extension topology, because the version from Steen-Seebach is less general than what's on Wikipedia (from what I saw, Steen-Seebach just defines these extensions by adding a single point).

I occasionally see the words "non-Hausdorff cone". No idea what that is, but would that be related to this construction?

Unfortunately for S48's description / name, the non-Hausdorff cone from the finite topology sources is the open extension topology from Steen-Seebach. The "non-Hausdorff cone point" is a focal point, not a generic point.

[prabau]

I think it's fine to have the primary name in non-algebraic terms in this case, and we also have $\mathrm{Spec}(\mathbb Z)$ for the algebraic description. Both of these precisely identify the space. The other alias (Prime ideal topology) is the least helpful, but is there because of S&S. I think it should move to the last position.

Also when referring to a wikipedia entry, the name in the refs: section is usually of the form "... on Wikipedia".

[prabau]

The name "Closed extension topology" is not used much in the literature, but it does appear in a few places, mainly due to references to S&S. Not sure there is a better way to express things.

[prabau]

Should the P201 trait be described in terms of the extra point, now that we emphasize the more topological description? ("Every nonempty open set contains the point $p$.")

[GeoffreySangston]

The name "Closed extension topology" is not used much in the literature, but it does appear in a few places, mainly due to references to S&S. Not sure there is a better way to express things.

Yea this is the issue I found. The most famous representative for this homeomorphism class of spaces is clearly (the underlying space of) $\mathrm{Spec}(\mathbb{Z})$. But it seems like the best description of the space from the point of view of general topology is the direct unwrapping of the definition of the closed extension topology applied to S15. I would guess this name (the closed extension one) is somewhat opaque to most passersby, which is unfortunate. It does have the strength of conveying to the reader that it has something to do with another space that they probably are familiar with. The problem with ' $\mathrm{Spec}(\mathbb{Z})$ ' (or 'Prime Ideal Topology') as the primary name is it sends the reader off into another subject just for one example. (It's not clear if these latter two are less opaque than the closed extension one, except for algebraic geometers. Still, ' $\mathrm{Spec}(\mathbb{Z})$ ' likely the most recognizable.)

Should the P201 trait be described in terms of the extra point, now that we emphasize the more topological description? ("Every nonempty open set contains the point p .")

I was thinking about updating all of the traits (or as many as is convenient) to reflect the new definition. Would it be good to do that here? I'll go ahead and change P201 now.

[prabau]

Yeah, although not ideal, "Close extension of ..." is probably as good as any. Unless someone comes up with a better suggestion, let's go with that one.

Up to you if you want to update more traits now, or leave it for a later time.

[GeoffreySangston]

Up to you if you want to update more traits now, or leave it for a later time.

I think it's good to open this up to other contributors, since I think a few other people wanted to make changes to S48 since this PR appeared. I'll be editing the traits to use the new language over the break though.

(I do want to clarify, and I think we agree, that one of the justifications for this change is that it should be easier to directly assert the conditions going into 'Spectral' now, and that I think 'Spectral' should not be asserted as a trait.)