The property "cannot be written as a countable disjoint union of closed subsets with at least two being nonempty"

[Jianing-Song]

The property "if $X=\displaystyle\bigcup_{n\in\mathbb{N}} C_n$ with $(C_n)$ being a countable disjoint family of closed subsets, then one of $C_n$ is $X$ and the others are empty" is quite interesting. Equivalently, this property can be stated as "every continuous map from $X$ to $\mathbb{N}$ equipped with the cofinite topology is constant". The property is stronger than connectedness.

We then have the theorems:
(i) compact + connected + Hausdorff $\Rightarrow$ this property (see https://math.stackexchange.com/q/6314);
(ii) locally compact + connected + locally connected + Hausdorff $\Rightarrow$ this property (see https://doi.org/10.4153/CMB-1973-069-1).

[StevenClontz]

We probably want a slicker title for it, but the existence of https://doi.org/10.4153/CMB-1973-069-1 makes me inclined to accept a contribution that introduces the property to pi-Base.

There's already a lot proposed here - I think the first step is to add the property, a theorem that says this property implies P36 connected, and at least one counterexample of a connected space without this property.

Then future contributions can add the theorems (i) and (ii).

[prabau]

Yeah, it seems it would be interesting to add the property, as there are various results in the literature mentioning it, even if not with a special name.

@Jianing-Song We are always looking for new contributors. If you are interested, you can write some PR (pull request) and we can review it and get this into pi-base. If you have not done that before on another github project, we can guide you through it. Let us know. (And best to do in multiple PRs following Steven's suggestion above, so you get the hang of it, and it's easier to manage the release into pi-base.)

Regarding the name, not sure if there is a good name in the literature somewhere.

"Cannot be written as a countable disjoint union of closed subsets with at least two being nonempty" is perfectly descriptive, but unwieldy.
"Not a countable disjoint union of closed sets" is shorter and easier to manage (from the title of the referenced paper in Canadian Math. Bulletin). It is not quite right, as one has to understand in the definition that we must exclude the case of all sets empty except one. But these assumptions are the only ones that make sense to avoid trivialities in such a definition. So as long as it's explained correctly in the text itself, I think such a name would be fine. But feel free to propose other suggestions.

Note that "countable" means finite or countably infinite, like in the rest of pi-base. The case of finite means such a space is automatically connected. (That's one simple theorem linking this property to existing pi-base properties.)

Question: What about a finite connected set? It seems to match the definition. No problem with that even if it's a degenerate case in a way, right?

[Jianing-Song]

Yeah, it seems it would be interesting to add the property, as there are various results in the literature mentioning it, even if not with a special name.

@Jianing-Song We are always looking for new contributors. If you are interested, you can write some PR (pull request) and we can review it and get this into pi-base. If you have not done that before on another github project, we can guide you through it. Let us know. (And best to do in multiple PRs following Steven's suggestion above, so you get the hang of it, and it's easier to manage the release into pi-base.)

Regarding the name, not sure if there is a good name in the literature somewhere.

"Cannot be written as a countable disjoint union of closed subsets with at least two being nonempty" is perfectly descriptive, but unwieldy. "Not a countable disjoint union of closed sets" is shorter and easier to manage (from the title of the referenced paper in Canadian Math. Bulletin). It is not quite right, as one has to understand in the definition that we must exclude the case of all sets empty except one. But these assumptions are the only ones that make sense to avoid trivialities in such a definition. So as long as it's explained correctly in the text itself, I think such a name would be fine. But feel free to propose other suggestions.

Note that "countable" means finite or countably infinite, like in the rest of pi-base. The case of finite means such a space is automatically connected. (That's one simple theorem linking this property to existing pi-base properties.)

Question: What about a finite connected set? It seems to match the definition. No problem with that even if it's a degenerate case in a way, right?

Thanks! Yes I am new to github, so I have never done a pull request before, but I would be happy to learn to make one.

As for the name, the problem is a "finite connected space" seems to be a topo space that is finite and connected (this is what I got when I searched this phrase in google). I personally did not come up with a good enough name either; the shortest that came into mind is "Has no nontrivial countable partition of closed sets".

[prabau]

Sorry, it seems I misrepresented what I wanted to say. I was just noting that spaces that are finite and connected would match the definition of the new property. And thinking out loud that there would be no problem with this degenerate case.

• (1) Cannot be written as a countable disjoint union of closed subsets with at least two being nonempty
• (2) Has no nontrivial countable partition by closed sets
• (3) Not a countable disjoint union of closed sets

I like your suggestion (2) or (3) from the article title. We can discuss more when reviewing the PR.

[prabau]

Thanks! Yes I am new to github, so I have never done a pull request before, but I would be happy to learn to make one.

Awesome!

What I would suggest is to first become familiar with the high level ideas of git and github: https://docs.github.com/en/get-started.
Specifically look at About GitHub and Git and go through the Hello World tutorial to get familiar with the ideas of git repository, commit and branch, as pi-base uses this on a daily basis. But no need to get bogged down into too many details. Just the high level should be enough to start with.

Once you get through those, you can clone the pi-base repository and create a PR on your local repo. We can discuss this in more detail when you get to that point. Just reach out.

@StevenClontz can give you more pointers too.

I have assigned this issue to you. But take your time, no rush.

[StevenClontz]

In addition to asking any specific questions you like here, I've written a guidebook at https://g4m.clontz.org that may be helpful.

There's also https://github.com/pi-base/data/wiki/Contributing which has more specific guidance for this project.

[Jianing-Song]

Thanks! Yes I am new to github, so I have never done a pull request before, but I would be happy to learn to make one.

Awesome!

What I would suggest is to first become familiar with the high level ideas of git and github: https://docs.github.com/en/get-started. Specifically look at About GitHub and Git and go through the Hello World tutorial to get familiar with the ideas of git repository, commit and branch, as pi-base uses this on a daily basis. But no need to get bogged down into too many details. Just the high level should be enough to start with.

Once you get through those, you can clone the pi-base repository and create a PR on your local repo. We can discuss this in more detail when you get to that point. Just reach out.

@StevenClontz can give you more pointers too.

I have assigned this issue to you. But take your time, no rush.

Thanks! I will look at these resources in my free time.

[Jianing-Song]

Thanks! Yes I am new to github, so I have never done a pull request before, but I would be happy to learn to make one.

Awesome!

What I would suggest is to first become familiar with the high level ideas of git and github: https://docs.github.com/en/get-started. Specifically look at About GitHub and Git and go through the Hello World tutorial to get familiar with the ideas of git repository, commit and branch, as pi-base uses this on a daily basis. But no need to get bogged down into too many details. Just the high level should be enough to start with.

Once you get through those, you can clone the pi-base repository and create a PR on your local repo. We can discuss this in more detail when you get to that point. Just reach out.

@StevenClontz can give you more pointers too.

I have assigned this issue to you. But take your time, no rush.

I've made the edits in my own repository. Before making the corresponding pull request, I would like the chance to know if it is possible to preview the edits (as if they had already taken effect)? Thanks!

[StevenClontz]

The easiest way is for me to make a copy of your branch on this repo. Make a draft PR and we can make that happen

[Jianing-Song]

Thanks! Yes I am new to github, so I have never done a pull request before, but I would be happy to learn to make one.

Awesome!

What I would suggest is to first become familiar with the high level ideas of git and github: https://docs.github.com/en/get-started. Specifically look at About GitHub and Git and go through the Hello World tutorial to get familiar with the ideas of git repository, commit and branch, as pi-base uses this on a daily basis. But no need to get bogged down into too many details. Just the high level should be enough to start with.

Once you get through those, you can clone the pi-base repository and create a PR on your local repo. We can discuss this in more detail when you get to that point. Just reach out.

@StevenClontz can give you more pointers too.

I have assigned this issue to you. But take your time, no rush.

Just another thing: In the Math Stack Exchange question cited above (https://math.stackexchange.com/q/6314) it is stated that the proof of "compact + connected + Hausdorff $\Rightarrow$ this property" can be found in General Topology by Ryszard Engelking (Thm 6.1.27). Could you please give me a hint on how to cite it?

[prabau]

Just another thing: In the Math Stack Exchange question cited above (https://math.stackexchange.com/q/6314) it is stated that ... can be found in General Topology by Ryszard Engelking (Thm 6.1.27). Could you please give me a hint on how to cite it?

You can look at T384 (and many other places) for an example. In the refs: section you add
- mr: 1039321
name: General Topology (Engelking, 1989)

and in the text, something like this for example:
See Theorem 6.1.27 of {{mr:1039321}}.

[Jianing-Song]

The easiest way is for me to make a copy of your branch on this repo. Make a draft PR and we can make that happen

Thanks! I've made a pull request. Please let me know if there is something wrong or imprecise!

[Jianing-Song]

Just another thing: In the Math Stack Exchange question cited above (https://math.stackexchange.com/q/6314) it is stated that ... can be found in General Topology by Ryszard Engelking (Thm 6.1.27). Could you please give me a hint on how to cite it?

You can look at T384 (and many other places) for an example. In the refs: section you add - mr: 1039321 name: General Topology (Engelking, 1989)

and in the text, something like this for example: See Theorem 6.1.27 of {{mr:1039321}}.

Thanks for you help! The MR number is exactly what I was looking for.

[prabau]

The MR number is exactly what I was looking for.

FYI, github.dev (vscode editor) has a search feature. I used that to find examples with "Engelking".

[prabau]

Also, you can edit the description of the PR you created to reference this particular issue (number preceded by #). Something like See issue #593. And as a result, github will automatically create links between the two, which is pretty handy.

[Jianing-Song]

The MR number is exactly what I was looking for.

FYI, github.dev (vscode editor) has a search feature. I used that to find examples with "Engelking".

Also, you can edit the description of the PR you created to reference this particular issue (number preceded by #). Something like See issue #593. And as a result, github will automatically create links between the two, which is pretty handy.

Thanks a lot! I will remind that.

[Moniker1998]

A space with this property is called $\sigma$-connected, see Continuum theory by Nadler, chapter 5 section 4

Every continuum admits the stronger property of being $\sigma$-connected, a theorem by Sierpiński

[prabau]

@Moniker1998 Good to know about that terminology. I don't know if Nadler uses standard terminology or makes up his own. Would you know of some recent papers that use $\sigma$-connected?

[prabau]

Found one. Not so recent, but it fits:

https://www.impan.pl/en/publishing-house/journals-and-series/colloquium-mathematicum/all/41/1/102462/hereditarily-connected-continua

[Jianing-Song]

Ah, I shouldn't have closed this issue because we haven't add Nadler's locally compact, connected, Hausdorff but not σ-connected
example. But please forgive me, I probably have to first deal with the other issues opened by me :)