Property Suggestion: Weakly locally path connected

[felixpernegger]

Property Suggestion

A space is said to be weakly locally path connected if every point has a path connected (equivalently open) neighborhood. Equivalently, every path component of a space is open.

Rationale

This generalises a few theorems slightly (like connected + locally path connected => path connected) and can come in handy with the new weakly locally simply connected (P231) property.
Having this property occured to me reading the comment #1654 (comment)

If we have P231, I think it makes sense to have this also.

[mathmaster13]

As a matter of preference, I think "path components are open" is wayyy clearer, but I know we have "weakly locally X" and this is weakly locally PC.
It definitely would simplify some theorems, and provide a more "fundamental" reason why they work.

Example:
connected + path components are open -> path connected (replaced connected + locally PC -> PC)
path components are open + totally path disconnected -> discrete (replaces locally PC + totally PD -> discrete)
LPC -> path components are open
P231 -> path components are open
P232 -> path components are open
path connected -> path components are open (trivial but important)

Whether this justifies its inclusion, I can't say. But it is sometimes nice to think about. I was discussing nLab's strange definition of SLSC, for example, which ends up being SLSC(in our sense) plus this condition. It is the core of everyone's favorite counterexamples involving connected non-path connected spaces, because a connected space is (trivially) path-connected if and only if this condition holds. All in all, it is a useful gadget to have and to be able to query.

[mathmaster13]

Fun and useless fact: it is also equivalent to the seemingly weaker property "every point x in X has a neighborhood (eq. open) U so that the map $\pi_0(U) \to \pi_0(X)$ is trivial. In other words, it is the $\pi_0$ analog of SLSC. (Which actually makes nLab's definition very pretty, though I still believe we picked the right definition for pi-base). While we have no reason to add this name to pi-base, it is quite cool that this property is "semilocally path-connected".

[felixpernegger]

As a matter of preference, I think "path components are open" is wayyy clearer, but I know we have "weakly locally X" and this is weakly locally PC. It definitely would simplify some theorems, and provide a more "fundamental" reason why they work.

Example: connected + path components are open -> path connected (replaced connected + locally PC -> PC) path components are open + totally path disconnected -> discrete (replaces locally PC + totally PD -> discrete) LPC -> path components are open P231 -> path components are open P232 -> path components are open path connected -> path components are open (trivial but important)

Whether this justifies its inclusion, I can't say. But it is sometimes nice to think about. I was discussing nLab's strange definition of SLSC, for example, which ends up being SLSC(in our sense) plus this condition. It is the core of everyone's favorite counterexamples involving connected non-path connected spaces, because a connected space is (trivially) path-connected if and only if this condition holds. All in all, it is a useful gadget to have and to be able to query.

I agree with this. "Just" improving theorems for LPC is probably not enough but:

• There is a nice equivalent characterisation
• having P231 and not this would be a bit artificial

[prabau]

For comparison, note that P231 (weakly locally simply connected) does not require the nbhd to be open. That would have been the P_3 property from the mo post, which we chose not to have in pi-base. (I don't think we know whether P_3 and P_4 are equivalent.)

But for this new property, if I am not mistaken, the following are all equivalent:

• every point has a path connected nbhd.
• every point has an open path connected nbhd.
• path components of $X$ are open

The equivalence of the first two above follows from similar arguments as the ones showing the equivalences for P42 (locally path connected), and https://math.stackexchange.com/a/3002235 in particular.

I agree with you guys that it would be a convenient property to have, even if it has not been formally used in the literature. Has it?

[prabau]

Checking in google scholar, I see a few papers using "weakly locally path connected" to mean $LC^0$; equivalently, for every $x\in X$, every nbhd $U$ of $x$ contains a nbhd $V$ of $x$ such that every point of $V$ can be joined to $x$ by a path in $U$.
Again, it's also related to "$X$ is path connected im kleinen at every point" (see https://en.wikipedia.org/wiki/Locally_connected_space). And all that ends up being equivalent to "locally path connected".

So it may be preferable to use a name like "Has open path components". We'll have to see.

[felixpernegger]

I also dont see it used anywhere (except in one lecture notes from Munich...), so "Has open path components" seems preferable, yeah

[prabau]

Searching for "path components are open" in google scholar does return some hits, where the property appears in various theorems.

Also,
Franklin & Smith-Thomas, Topologies determined by paths https://zbmath.org/0451.54017 (https://topology.nipissingu.ca/tp/reprints/v04/tp04204.pdf), where it's called PCGS.

And these lecture notes:
https://www.math.uni-bielefeld.de/~tcutler/pdf/Elementary%20Homotopy%20Theory.pdf

[felixpernegger]

I just make PR then I guess?

[prabau]

Sure. Please don't cram all the possible related theorems into it if there are too many. First a simple PR with the definition and a few basic theorems relating it to other properties. Then more of them.

[prabau]

See #1668 for a possible related property.

[felixpernegger]

Added in #1662