Property: Path components are open

[felixpernegger]

#1661

All theorems are added are almost trivial. Maybe one of T523 or T602 also generalises, I did not think about this. I dont know any other theorems. (T523 does generalise, but let's not do it here, in T602 "path connected" is equivalent t "weakly locally path connected" anyways already) I dont know other theorems.

For transparency, out of curiosity I tried using Claude Code to make the property page, but it didnt do so well.

[prabau]

"weakly locally path connected" is not used much in the literature, and when it's used, it has a different meaning. So we should not have it as an alias.

[felixpernegger]

"weakly locally path connected" is not used much in the literature, and when it's used, it has a different meaning. So we should not have it as an alias.

It's at least used 1 time here: https://www.mathematik.uni-muenchen.de/~land/Topologie1.pdf
and follows our conventions from the wiki, so I'd keep it personally.

[prabau]

It's at least used 1 time here: https://www.mathematik.uni-muenchen.de/~land/Topologie1.pdf
and follows our conventions from the wiki, so I'd keep it personally.

Right, but this is not published literature. It's a set of lecture notes, where the lecturer is free to come up with his own terminology. In published literature, the meaning of it is different and not compatible.

[prabau]

Let's ask opinion from other folks. @GeoffreySangston @mathmaster13 @StevenClontz

[mathmaster13]

Right, but this is not published literature. It's a set of lecture notes, where the lecturer is free to come up with his own terminology. In published literature, the meaning of it is different and not compatible.

Not sure if that's true. If a lecturer used nonstandard terminology, especially in an introductory course, it's basic good manners to mention that it's nonstandard, and to mention the standard terminology that your students will use for the rest of their lives.

Whether people do this or not is another question. He could have just been unaware of the other papers' existence, or used this term from somewhere he learned. I don't know.

[prabau]

Regarding all the variants of "locally path connected", there is no really "standard" terminology. And the property introduced here is weaker than the usual "locally path connected". So at least it seems a very reasonable choice to name it "weakly locally path connected" if one needs to talk about it. That would explain why the lecturer choose that terminology in the context of these lecture notes. No need to give more explanations than that in the particular set of lecture notes, which are not meant as a comprehensive reference with pointers to the literature either. That's how I would explain it.

[prabau]

I looked in more details at the use of "PCGS" and I don't see it as having been used anywhere in the literature, except in this paper of Franklin & Smith-Thomas, where it's used as an illustration of a more general notation $\mathcal A$ GS (local to this paper really). So I don't see a problem mentioning the paper, but I don't think it warrants adding PCGS to the list of aliases.

How about changing that paragraph to the following maybe:
This property is used in the literature, but usually without a separate name. One exception is {{zb:0451.54017}} (<https://topology.nipissingu.ca/tp/reprints/v04/tp04204.pdf>), where it is called "PCGS" (path connected generated space).

[StevenClontz]

"weakly locally path connected" is not used much in the literature, and when it's used, it has a different meaning. So we should not have it as an alias.

If we follow https://github.com/pi-base/data/wiki/Conventions-and-Style#local-properties then "weakly locally path connected" should mean "each point has a path connected neighborhood". If there is a distinct property in the literature referred to as such, we should disambiguate.

[GeoffreySangston]

Let's ask opinion from other folks. @GeoffreySangston

I always prefer more descriptive names, so I don't really like qualifiers like "strong" and "weak" for pi-base. To me "strongly" and "weakly" don't make sense out of some specific context where you're working with two or three properties. I think the idea of using "weakly" as a convention to mean "each point has a neighborhood which has property" is fairly common, but then it turns out that even this seems to have variants (like in the case of locally simply connected in the sense of Chevallay vs. Knapp). Since pi-base could potentially have many properties which are minor variations on each other, I'd prefer very explicit names. Hopefully there are people actually using pi-base has a reference, and I doubt they're aware of the convention in the wiki. I think the "weakly" convention out in the wild is probably not as precise as the one in the wiki.

If it's used repeatedly in the literature with that name though I'd say add it as an alias, and if it's not then I'd cast my vote to leave it off.

[StevenClontz]

jinx @GeoffreySangston

I agree "weakly" isn't semantic as an adjective -- one has to refer to https://github.com/pi-base/data/wiki/Conventions-and-Style#local-properties to know what we mean. We could just say "Has P neighborhoods" and "Has local bases of P neighborhoods" for each property "P" as the main name for each property, and then use "(weakly/strongly) locally P" based upon what pops up in the literature.

[prabau]

Why is the property preserved by arbitrary products?

[mathmaster13]

Why is the property preserved by arbitrary products?

I'd think it isn't. Take a countable product of two-point discrete spaces {0, 1}.. Each two point space has open path components.

But the product does not. Take a neighborhood of (0, 0, 0, 0, ...). It must contain {0}^n times {0, 1}^aleph0 for some n. Project onto the n + 1st coordinate and you get a surjection onto the two point discrete space and so you can't be path connected.

[felixpernegger]

Why is the property preserved by arbitrary products?

I'd think it isn't. Take a countable product of two-point discrete spaces {0, 1}.. Each two point space has open path components.

But the product does not. Take a neighborhood of (0, 0, 0, 0, ...). It must contain {0}^n times {0, 1}^aleph0 for some n. Project onto the n + 1st coordinate and you get a surjection onto the two point discrete space and so you can't be path connected.

yeah finite product, sorry

[felixpernegger]

Regarding all the variants of "locally path connected", there is no really "standard" terminology. And the property introduced here is weaker than the usual "locally path connected". So at least it seems a very reasonable choice to name it "weakly locally path connected" if one needs to talk about it. That would explain why the lecturer choose that terminology in the context of these lecture notes. No need to give more explanations than that in the particular set of lecture notes, which are not meant as a comprehensive reference with pointers to the literature either. That's how I would explain it.

For any property P on topological spaces, we have "local P" (every point of a space has a neighborhood basis satisfying P) and "weakly locally P" (every point has a neighborhood satisfying P). As far as I can see this convention isn't used too much outside of pi-base (+ related stuff), but certainly a little bit (as observed by the lecture notes).

This is also reflected in the wiki here https://github.com/pi-base/data/wiki/Conventions-and-Style#local-properties.

So even if this term (weakly locally path connected) has not seriously appeared in literature yet, it makes sense to at least include it as an alias in my opnion.

[prabau]

But when the term "weakly locally P" is used in the literature with a different meaning, including it as an alias is misleading. It seems preferable not to have it in this case. I think @GeoffreySangston agrees (#1662 (comment)). Not sure I quite understood Steven's recommendation in this case (#1662 (comment) and #1662 (comment)).

@StevenClontz can you clarify?

Furthermore, if we really want to have "weakly locally path connected" as an alias, we would need some accompanying text, something like:

"The terminology "weakly locally path connected" has appeared occasionally in the published literature with a different meaning, not compatible with the pi-base convention of using "weakly locally P" to indicate that each point has a neighborhood satisfying P."

Not sure what other people think about that, but it does not seem helpful to me to add this alias then. Thoughts?

[prabau]

Worse, we would even have to say:

"The terminology "weakly locally path connected" does not occur in the published literature with the current meaning. It has appeared occasionally with a different meaning, not compatible with the pi-base convention of using "weakly locally P" to indicate that each point has a neighborhood satisfying P."

[felixpernegger]

Whatever then. This is not so important....

[felixpernegger]

Worse, we would even have to say:

"The terminology "weakly locally path connected" does not occur in the published literature with the current meaning. It has appeared occasionally with a different meaning, not compatible with the pi-base convention of using "weakly locally P" to indicate that each point has a neighborhood satisfying P."

But won't making a systematic overview (like pi-base) inevitably have small overlaps like this? Obviously it would be a bad idea to "change" the definition of something common (like connected), but for something very niche like this property, I dont really see the issue.
We also don't mention that "Connected" as we have it often means "Connected or empty" or that "Compact" is often refered to "Quasicompact" (i.e. Compact = Quasicompact + T2) and I don't think there is a need for that.

[prabau]

I don't disagree at all.

All I was saying is that if we wanted to introduce "weakly locally path connected" as an alias, we should have explained more, given what was already discussed. And the resulting paragraph would have seemed kind of pointless. The difference with the examples you just mentioned is that those properties do appear in the literature with the meaning in pi-base.

[prabau]

Can you give the argument here for box products?

[felixpernegger]

Can you give the argument here for box products?

actually I just remove it. Not sure if box product of simply connected is simply connected

[prabau]

The countable box product of the reals is not connected and not path connected. From the answer of Jakobian to https://math.stackexchange.com/questions/5012753, it seems that its path components are not open either.