Property Suggestion: Locally contractible

[GeoffreySangston]

We should add one or more locally contractible properties, determine which variations of locally contractible are appropriate to add, and determine good names and aliases for these. We should also gather examples distinguishing the variations we add, and we may need to gather useful equivalent definitions. We should establish a minimal first PR which hooks up the basic connections once we've decided on the properties.

We recently added three variations on locally simply connected: P230, P231, and P232. There may be useful discussion in the issue thread for locally simply connected, which I'm modeling this post off of: #1654. My initial assumption was that we would at least add the direct modifications of current three variants of locally simply connected, though now I am not so sure. I'll list those now:

1. Locally contractible - $X$ admits a basis of open sets which are {P199}.

2. Weakly locally contractible - Every point of $X$ has a neighborhood which is {P199}.

3. $LC$ - $X$ is locally contractible at each of its points. A space $X$ is said to be locally contractible at a point $x_0 \in X$ provided that each neighborhood $U$ of $x_0$ contains a neighborhood $U_0$ of $x_0$ which is contractible in $U$ to a point. (Note: This definition is verbatim from Borsuk's book.)

Here is an older issue thread suggesting locally contractible: #1150. This thread lists variations with references, and tries to say when they are equivalent. Notably, neither the Locally contractible nor the Weakly locally contractible given above actually appear. This thread also has many references for a second property like $LC$ which requires the homotopies to fix the point (family 2).

I am confident we should add $LC$, and unsure about exactly which other properties should be added.
I think we should also consider adding definition 2', since that seems to appear often as well. Due to Fadell's paper (see below) showing definition 4 is a homotopy invariant, I think definition 4 could potentially be helpful for pi-base. I.e., contractible spaces automatically satisfy definition 4, but may not satisfy the others. More importantly since this would be the weakest definition, a technique for showing a space is not locally contractible in any sense would be to show it is homotopic to one not satisfying this.

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Potentially useful links:

• Terminology for local contractibility:..
• ANR is locally contractible
• Distinguishing locally contractible (in the sense of Borsuk) at $x$ and to $x$ for every $x$.

[felixpernegger]

Like in locally simply connected, an obvious question is whether "open neighborhood" can be taken instead of neighborhood.

I suppose we don't know this?

[GeoffreySangston]

@felixpernegger Right. I don't know the answer to that. I also don't have a best source for the open basis of contractible sets version (or the local base version). I did an exact Google Scholar search for "locally contractible" "basis of contractible", and there are 35 hits, but I'm not seeing a best reference. There are >3k hits in the exact search for "locally contractible". I also looked through the hits from <1965 and some saw interesting things, but nothing jumped out at me as worth sharing here.

An exact Google Scholar search for "strongly locally contractible" shows it being used in a variety of ways. Also see an exact Google Scholar search for "strong local contractibility". I'm seeing the local base version (the one on Wikipedia) used e.g. in one arbitrarily chosen paper here. The old thread lists a few sources using the strong terminology for definitions 2 and 2' of that thread.

[felixpernegger]

@felixpernegger Right. I don't know the answer to that. I also don't have a best source for the open basis of contractible sets version (or the local base version). I did an exact Google Scholar search for "locally contractible" "basis of contractible", and there are 35 hits, but I'm not seeing a best reference. There are >3k hits in the exact search for "locally contractible". I also looked through the hits from <1965 and some saw interesting things, but nothing jumped out at me as worth sharing here.

So just to make sure I understand correctly, the >3k results are mostly number 1. from your list above? I didnt know this property is so common.

To stay somewhat consistent, maybe we can add a version of locally contractible that mirror the ones we added for locally simply connected, so we get all the implications immediately? Just saw you adressed this already...
I also think its okay to still modify the locally simply connected ones, so everything fits better together (having 3 versions of essentially the same property may be a bit much anyways), but you clearly now best about this :)

[GeoffreySangston]

the >3k results are mostly number 1

I'm afraid of making any confident statements since I'm really not an expert on this, but I think both number 1 and definition 2' (Edit! I only now realized definition 2 and 2' are likely not equivalent. 2' is the common one) are very common. It seems hard to find the naive version of locally contractible.

I also think its okay to still modify the locally simply connected ones, so everything fits better together

I agree. I'd just re-emphasize my personal preference would be to avoid giving proper names to the properties that actually don't seem to occur in published literature, but stick with descriptive titles (maybe "Points have contractible neighborhoods" and "Has an open basis of contractible sets"). This seems likely to lead to less confusion for anyone learning from pi-base. I also emphasize that "weakly locally contractible" or "weak local contractibility" seem to be quite common aliases for number 1 and possibly 2 of the list of definitions, which doesn't correspond with our current weakly locally simply connected. This article uses this and refers to the local base version as 'strongly locally contractible' (and does not seem to define 'locally contractible' but uses it generically?).

[GeoffreySangston]

This 1959 article by Fadell studies the definition of weakly locally contractible used in Dugundji (item 4 from the old thread). Fadell calls it weakly locally contractible and justifies using it over "the perhaps more natural property "locally contractible"", because he shows it is a homotopy invariant. I'm not sure now what the more natural one is supposed to be. It is interesting this version is a homotopy invariant because definition 1 (Borsuk's) from the old thread (and I think 2' is not -- probably same for the local base and the open basis ones) π-Base, Search for contractible + ~Locally path connected. Borsuk's is of course invariant under retractions. I guess this is probably true of definition 2' from the old thread.

[prabau]

Regarding the Dugundji/Fadell definition: A space $X$ "weakly locally contractible" if each $x \in X$ has a nbd $U$ deformable over $X$ to $x$." (Deformable to $x$ means the existence of a homotopy from the identity map to the constant map with value $x$.

(Aside: it's equivalent to just ask that the nbd $U$ be deformable to some point (i.e., say that the inclusion $U\hookrightarrow X$ is null-homotopic): if the point is not $x$, we can follow the first homotopy with a path from that point to $x$.)

This is not really a "local" property because the homotopy is allowed to occur over the whole space $X$. It is more a "semilocal" property. In fact, Sakai, Geometric aspects of general topology (p. 349) calls it exactly that: $X$ is semilocally contractible if "each point of X has a neighborhood that is contractible in X".
That was also noted by user Dennis as a comment to https://math.stackexchange.com/questions/5109428.

I think this would be a good property to add as it has an obvious relation with "semilocally simply connected". In a separate PR from the others maybe. Would be good to find examples to discriminate from other properties.

As for the terminology, Google scholar returns hits for "semilocally contractible" and "semi-locally contractible". Unfortunately, some of them use the terminology to mean Borsuk's LC, or something else. I did not look at more than a handful of links though. Would have to check more thoroughly.