Adds contractible = true to spaces with excluded/particular point topologies

[GeoffreySangston]

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[GeoffreySangston]

I asked on MSE if a general condition which would simultaneously cover the particular + excluded point topologies is true.

Update: David Gao wrote a really nice proof. David's proof is really a proof of 'X path connected + has dispersion point => (X contains a dense point or X contains a point which is contained in no proper open subset of X)'. Then Lemma 2.3.2 from Peter May's book draft applies. David Gao also gives a proof of this lemma. I think David Gao's proof of the lemma is actually more explicit just because Peter May's book says the argument is the same in the two cases, and there appears to be a subtle difference (which does seem interesting to me).

Instead of adding 'path connected + has a dispersion point => contractible', I think I would like to factor the properties a bit. This is just because the assumptions going into Peter May's Lemma naturally lead to the contraction maps, and these assumptions seem natural in their own right. I'm thinking of making the following changes.

1. Remove all of the files I added in the commit.
2. Add 'Has a dense point' as a property to the pi-Base. (I.e., there exists a point p in X such that the closure of {p} is X.)
3. Add 'Has a dense point' to each of the particular point topologies.
4. Add 'Has a point contained in no proper open subset' as a property to the pi-Base. I don't like this name though and don't know what a better name would be. Maybe somebody could provide a suggestion?
5. Add the property from 4 to to each of the excluded point topologies.
6. Add 'Has a dense point => contractible', using the construction at the bottom of David Gao's post, and also linking to it. I might also link to Peter May's book.
7. Add '(Has a point contained in no proper open subset) => contractible', using the construction at the bottom of David Gao's post, and also linking to it. I might also link to Peter May's book.

I don't think pi-Base supports anything like dynamically constructing the union of conditions as a new property. So I suppose to get David Gao's proof in the pi-Base I have to do one of the following. The following is iffy. Does anybody have a better idea than either of these two possibilities?
8.
Possibility A:
A1. Add '~(Has a point contained in no proper open subset) + Has a dispersion point + path connected => Has a dense point'. Link to David Gao's post.
A2. Add '~(Has a dense point) + Has a dispersion point + path connected => Has a point contained in no proper open subset'. Link to David Gao's post.

Possibility B:
B1. Add '(Has a point contained in no proper open subset) or (Has a dense point)' as a property to the pi-Base.
B2. Add 'Has a dispersion point + path connected => (the property add in B1)'
B3. Add the theorems connecting the property from B1 to
(Has a point contained in no proper open subset) and (Has a dense point).

In a future pull-request (or maybe a future commit to this one):
9. Add 'Zariski topology of Spec(Z)' as a new space to the pi-Base. This answer says the proper closed sets are any finite set which excludes the zero ideal, i.e., the proper open sets are the cofinite (Edit) sets which include the zero ideal. So this is like the particular point topology on a countably infinite set, but I think distinct from it. I need to learn more about Spec(Z) before writing this though.
10. Add 'Has a dense point' to 'Zariski topology of Spec(Z)', which is true according to nlab.

[StevenClontz]

"Has a dense point" seems like a fine utility property.

It's kind of made up, but define a global limit point to mean "a point which belongs to the closure of every non-empty set", equivalently, "a point that belongs to no proper open set". Then we could add "Has a global limit point".

I know @jamesdabbs has some plumbing to support more elaborate logic than AND(properties) => property; for example, we used to have several P+Q => R+S theorems, until we split them up for editorial reasons. And as you note, P+Q => R or S can be expressed as two theorems. I don't have a great idea of the best model right now.

[prabau]

FYI, a "dense point" is usually called a "generic point" of the space.

[prabau]

I need to read in detail what you wrote. But FYI, see also this answer from Eric Wofsey:
https://math.stackexchange.com/questions/2965227/are-minimal-neighborhoods-in-an-alexandrov-topology-path-connected

showing that a space with a point satisfying a similar condition is contractible. And similarly for the dual condition.

It's actually the same as David Gao's conditions.

[prabau]

Item 2: Adding "Has a generic point" seems reasonable to me. It's used in the literature, and there are connections to the notion of quasi-sober (obvious theorem to add).

I don't understand your suggestions A1, A2.
Please double check me on this, but the following are always equivalent for a point $p\in X$: (redundant, but just for clarity)

• $p$ is in every nonempty open set in $X$
• $p$ is contained in no proper closed set
• $X$ is the only closed set containing $p$
• The closure of $\{p\}$ is $X$
• $\{p\}$ is dense in $X$
• $p$ is a generic point of $X$

And also these are equivalent:

• $p$ is in every nonempty closed set in $X$
• $p$ is contained in no proper open set
• $X$ is the only open set containing $p$
• $X$ is the only neighborhood of $p$
• NEED TO GOOD TERMINOLOGY HERE

For the second batch, I don't know what a good terminology would be. But repeating what @StevenClontz has said multiple times, I don't think we want to introduce terminology of our own here. (see several comments to @Moniker1998 recently). Most probably somebody has come up with something in the literature already. And preferably it should be justified by simplifications or extensions of pi-base theorems and traits. (I guess the discussion above kind of does it. There are more examples than the excluded point topologies).

[prabau]

Items 9 and 10: Please don't add any Zariski topology stuff as part of this. It's a lot more involved and would deserve its own PR at the least.

[GeoffreySangston]

Item 2: Adding "Has a generic point" seems reasonable to me. It's used in the literature, and there are connections to the notion of quasi-sober (obvious theorem to add).

What do you think about 'Has a dense point' as an alias? I was influenced by this answer which says, "such a point is usually called a dense point", and which also says, "The related term generic point is used in algebraic geometry [sic] even when there is not an actual point that is in every open set, so to be unambiguous I would stick to dense point." I can't actually confirm if they're correct, since they don't specify where this is called a dense point. I could look around Google scholar.

I don't understand your suggestions A1, A2.

A and B are just alternative ways of getting a proposition of the form X => Y or Z into pi Base. A is to add both [X and ~Y => Z] and [X and ~Z => Y]. B is to create a new property named "Y or Z".

Please double check me on this, but the following are always equivalent for a point p ∈ X : (redundant, but just for clarity)

These lists look good. I'll likely use them to form the lists of equivalent definitions when I add the two new properties.

For the second batch, I don't know what a good terminology would be. But repeating what @StevenClontz has said multiple times, I don't think we want to introduce terminology of our own here.

I'll look around Google scholar.

[prabau]

What do you think about 'Has a dense point' as an alias? I was influenced by this answer which says, "such a point is usually called a dense point", and which also says, "The related term generic point is used in algebraic geometry [sic] even when there is not an actual point that is in every open set, so to be unambiguous I would stick to dense point." I can't actually confirm if they're correct, since they don't specify where this is called a dense point. I could look around Google scholar.

Maybe, maybe not. Mathse is not necessarily an authoritative source. Anybody can write anything there. Like you said, we should look at google scholar, as well as zbmath and maybe Encyclopedia of general topology.

I also don't understand what that guy says. A generic point $p$ is a point such that the closure of the singleton $\{p\}$ is the whole space. In other words, $\{p\}$ is dense in $X$. That exactly means that $p$ is in every nonempty open set.
Why does he say there may not be an actual point that is in every open set?

[GeoffreySangston]

@prabau

• NEED TO GOOD TERMINOLOGY HERE

Currently working on this problem. Barmak and Peter Mays' books on finite topology define a preorder (reflexive + transitive relation, possibly not anti-symmetric) associated to a finite (more generally Alexandrov) space by $x \le y$ if and only if $x$ is contained in every open neighborhood of $y$.

The Wikipedia page on specialization preorder defines the same concept (look down the page at the second definition after "For the sake of consistency") for arbitrary topological spaces. This page says there is disagreement on which direction the order goes in, and this page's convention differs from the two books mentioned. This page happens also to have a part about generic points, where it gives an explanation for the the etymology of 'specialization' and 'generic'.

Using the convention from Barmak, a maximum point under this preorder would be a point $y$ such that every $x \in X$ is contained in every open neighborhood of $y$. I.e., $X$ is the only open neighborhood of $y$. And a minimum point $z$ would be a point such that $z$ is contained in every non-empty open set (the first equivalent version of generic point you give above).

So I believe names / additional aliases for these properties could be "Has maximum point under the specialization preorder", "Has minimum point under the specialization preorder". We would need to decide which direction the preorder should go in, however. Alternatively, and probably better, we use "Has a generic point", and "Has a specialization preorder maximum point" (or minimum, depending on the best convention).

[prabau]

Given a preorder, there is a corresponding Alexandrov topology where the open sets are the upper sets, or are the lower sets, depending on your conventions. I think in pi-base we have been using more the convention that the open sets are the upper sets, which is also used in https://en.wikipedia.org/wiki/Alexandrov_topology and https://en.wikipedia.org/wiki/Specialization_(pre)order. But as you mentioned, I think people in algebraic geometry are more used to the opposite convention.

In any case, the notion " $x$ is a specialization of $y$" (notation: $y\rightsquigarrow x$) is independent of this choice for the Alexandrov topology (and makes sense in any space, even non-Alexandrov spaces). So instead of max or min for the specialization preorder, we can say something like " $x$ is a specialization of every other point".

Possible names for the property of the space:

"Has a point more specialized than every other point"
"Has a point that is a specialization of every other point"
"Has a point with $X$ as only neighborhood"
"Has a point contained in all nonempty closed sets"
"Has a point contained in all closed sets" (ignoring "nonempty", which only needs to be in the text)
etc, etc.

These are all paraphrases for what we are trying to say.

Is there an article somewhere that makes use of that notion, so we can see what they did?