Property Suggestion: locally contractible

[GeoffreySangston]

Property Suggestion

Primed versions are supposed to be equivalent.

1. (locally contractible in the sense of Borsuk.) For every $x \in X$, each neighborhood $U$ of $x$ contains a (not necessarily open) neighborhood $V$ of $x$ such that the inclusion map $V \hookrightarrow U$ is homotopic to a constant map (which does not necessarily have value $x$). The pointwise property is called locally contractible at a point $x \in X$.

• Borsuk's Theory of Retracts (Pages 28-30). Also uses LC as an alias.
• Encyclopedia of General Topology (EGT) (Page 341). Also uses LC as an alias.
• Bredon in Topology and Geometry (Page 536).
• Cappell-Ranicki-Rosenberg in Surveys on Surgery Theory : Volume 1 (Page 326)
• Wikipedia

1'. For every $x \in X$, each neighborhood $U$ of $x$ contains a (not necessarily open) neighborhood $V$ of $x$ such that the inclusion map $V \hookrightarrow U$ is homotopic to a constant map (which does not necessarily have value $x$). The pointwise property is called locally contractible to the point $x \in X$.

• Fuchs-Viro in Topology II: Homotopy and Homology. Classical Manifolds. (Page 14)
• Spanier. (Page 57)

1''. For every $x \in X$, each neighborhood $U$ of $x$ contains an open neighborhood $V$ of $x$ such that the inclusion map $V \hookrightarrow U$ is homotopic to the constant map with value $x$.

• Rotman's An Introduction to Algebraic Topology. (Page 211)

2. A space is locally contractible at a point $x \in X$ if every neighborhood $U$ of $x$ contains a neighborhood $V$ of $x$ such that $\{x\}$ is a strong deformation retraction of $V$.

• Lundell-Weingram in The Topology of CW Complexes. (Page 67)

(Edit from 2026: Looking back at this, I don't think this is equivalent to 2 above! This is the one which seems more important in terms of use in the literature however.). 2'. A space is locally contractible at a point $x \in X$ if every neighborhood $U$ of $x$ contains a neighborhood $V$ of $x$ such that the inclusion $V \hookrightarrow U$ is homotopic to the constant map with value $x$, such that the homotopy fixes $x$.

• Tom Dieck's Algebraic Topology. (Page 449)
• Banakh-Mine-Sakai-Yagasaki in Spaces of maps into topological group with the Whitney topology call this strongly locally contractible at $x$. Though this paper is not a general reference like the other sources so maybe should not be mentioned.
• Fuchs-Viro in Topology II: Homotopy and Homology. Classical Manifolds. (Page 15) calls this 'strong local contractibility'.

3. Wikipedia uses strongly locally contractible for 'Every point of $X$ has a local base of contractible neighborhoods'.

Wikipedia has the following:

Strong local contractibility is a strictly stronger property than local contractibility; the counterexamples are sophisticated, the first being given by Borsuk and Mazurkiewicz in their paper Sur les rétractes absolus indécomposables, C.R.. Acad. Sci. Paris 199 (1934), 110-112).

• I don't think 'strong local contractibility' is a standard name for this. The chapter by Fuchs-Viro in Topology II uses " strong local contractibility of X" for 2'' above. Looking at the Wikipedia edit dates, I suspect that somebody read the comment thread under this MSE answer and then took Moishe Kohan's suggestion that it is better to use "strongly locally contractible" for 3 (compared to 1), though I doubt Moishe Kohan meant for that to appear on the Wikipedia page; the Wikipedia edits happen some months after the the MSE comments.

4. Dugundji has an exercise which defines weakly locally contractible. "A space $Y$ is called weakly locally contractible if each $y \in Y$ has a nbd $U$ deformable over $Y$ to $y$." (Deformable to $y$ means the existence of a homotopy from the identity map to the constant map with value $y$. I've seen 'deformable' with this usage in a few places.)

(Hatcher is not mentioned because while he uses 'locally contractible' + 'locally contractible in the weak sense', I see no clear definitions of these in his book; Wikipedia states he uses 'weakly locally contractible', but I don't see this in the latest edition. I only mentioned sources which explicitly define the concept.)

Rationale

Making progress on #818.

Relationship to other properties

I'm going to assume only 1 will be added, at least for now. So this is specifically about 1.

1. Locally contractible => Locally path connected. (Will eventually be upgraded to Locally contractible => Locally simply connected.)
2. Locally Euclidean => Locally contractible
3. Based on the classification in the MSE answer linked from T523, T523 should be improved to 'LOTS + Path connected => Locally contractible'.
4. T316 should be upgraded to 'Alexandrov => Locally contractible', because each minimal neighborhood has a focal point for it, as mentioned in a MSE post.
5. Locally contractible + Has a focal point => Contractible. (The idea from New theorem: has a focal point plus a local property implies the global property #1112 (comment)). I'm not sure if it will ultimately be non-redundant.

Surely there are others, but I'm out of steam right now.

[GeoffreySangston]

Does everyone agree that 1 is the only game in town and should be what's called 'locally contractible' in pi-base? Does anyone know of a source that uses 2 as the main definition?

[StevenClontz]

Calling 1 "locally contractible" and not "weakly locally contractible" would require us to change our style guide you noted at https://github.com/pi-base/data/wiki/Conventions-and-Style#local-properties

I'm not famliar enough with the literature here to have a strong opinion for this specific property, but would want to either stick to the set convention, or have a discussion (it's been four years) whether the convention is ill-advised (here or in general).

[StevenClontz]

See also this open question: https://math.stackexchange.com/questions/4004934/definitions-of-locally-contractible-spaces

[GeoffreySangston]

Do you think a question on mathoverflow asking for authoritative references which use a different version of locally contractible from Borsuk's is the kind of thing that's going to be received well? I.e.,

I'm looking into the various definitions of 'locally contractible' which are explicitly defined in published authoritative texts. The version which appears in Borsuk's Theory of Retracts is.. . This differs from the way topologists define the local version of a given property (also see the recent question https://mathoverflow.net/q/484764/109654). Wikipedia claims that there is disagreement over what the standard definition of 'locally contractible' is, but I have not found evidence that there is disagreement in published authoritative texts which explicitly define the term; note that no explicit definition appears in Hatcher's algebraic topology book. E.g., algebraic topology texts by Rotman, tom Dieck, and Spanier use Borsuk's definition. Also Encyclopedia of General Topology follows Borsuk, and so does the chapter in Topology II (editors Novikov-Rokhlin) by Fuchs-Viro. As does Surveys on Surgery Theory : Volume 1 by Cappell-Ranicki-Rosenberg, Topology and Geometry by Bredon, and The Topology of CW Complexes by Lundell-Weingram. I gave up looking at this point. The one authoritative reference I have found which defines it differently is the online encyclopedia nlab, though no reference is given from that page; see locally contractible at nlab.

Is there a body of literature which uses a different definition than the one from Borsuk's textbook?

(I would want to double check all of these claims before asking it, of course.)

If there is a body of literature using a definition like one of the one's from your post, then we can cite that in pi-base's page for that property, and perhaps call the other one 'locally contractible in the sense of Borsuk' (I really need to confirm Borsuk is the originator), or perhaps 'locally contractible at every point'. Otherwise, I think there's a good case here to deviate from pi-base convention, and mention a deviation has occurred in the description.

(Also I was flattered you said I pointed something out in your mathoverflow question, but it was a genuine question :-). )

[prabau]

I did not read all you wrote above, but the word "nlab" caught my eye. In my opinion, nlab is not necessarily authoritative for terminology. For category theory, they know what they are talking about. But I have seen some cases for topology where their terminology was completely wrong. Explanation: just like wikipedia, people can come and add whatever they think is right, and it was one guy who added one article with incorrect stuff. (I would have fixed it, but their editing interface is so cumbersome ... sigh)

[prabau]

(For comparison) See https://en.wikipedia.org/wiki/Locally_connected_space

For locally connectedness at a single point $x$, there are two notions:

• locally connected at $x$
• connected im kleinen at $x$ (= weakly locally connected at $x$)

The difference having to do with open nbhds vs. not necessarily open nbhds.
And then the theorem that if a space is connected im kleinen at each point, it is in fact locally connected at each point.

Similarly, from the mathse post mentioned by Steven above, there is local contractibility at a point, phrased in terms of open nbhds or not, and also the definition from Paul Frost in the related link.
Would there be a corresponding global theorem saying it a space is "locally contractible im kleinen" at each point, then ... ?

[GeoffreySangston]

@prabau On that mathse page the definition cited to nlab seemed ambiguous since nlab uses the notation $U \hookrightarrow X$ (though to be fair I guess nlab is just calling the subspace $U$ contractible, so it does agree with mathse), and the one cited to Wikipedia is not similar to what appears on Wikipedia. It seems to be what Wikipedia calls strongly locally contractible, though I'm not sure that usage is common outside of Wikipedia.

I'm not sure about locally contractible im kleinen at x. I can think of multiple definitions of what that should mean, so it's not exactly clear what the question is. E.g.,

(Edit: Note that in Borsuk's locally contractible, every neighborhood $U$ of $x$ contains a neighborhood $U_0$ which is deformable in $U$ to a point, but not apparently not necessarily $x$! So I've edited that in the following.)

1. (In the sense of Borsuk.) $X$ is locally contractible im kleinen at $x$ if every neighborhood $U$ of $x$ contains a neighborhood $V$ of $x$ such that $V$ is deformable in $U$ to a point.

• Now that I double check, this is actually what Encyclopedia of General topology uses. Though I'm pretty sure Borsuk is their main source so I have to double check everything. Edit: This is actually exactly what Borsuk calls locally contractible at $x$.

2. (In the sense of a weak local property) $X$ is locally contractible im kleinen at $x$ if every neighborhood $U$ of $x$ contains a neighborhood $V$ of $x$ which is contractible as a subspace.

The issue is that Borsuk's definition of locally contractible is not (obviously at least?) exactly about some subspace being contractible.

[GeoffreySangston]

@prabau So Borsuk actually apparently the locally contractible im kleinen version. I'll have to separate out the sources I listed under 1. I'll do that when I get home.

[prabau]

Thanks. I have not thought much about any of that, just giving some food for thought about the different possibilities one could use. And possibly (or not) introduce multiple variants to illustrate the differences. It looks like it's not a small project.

[prabau]

"locally contractible im kleinen" is something I made up by analogy.

The better comparison would be with "locally path connected at a point" vs. "locally path connected im kleinen at a point",
with the difference that the local paths may need to leave the inner nbhd $V$ while remaining in $U$, or not.
(see https://en.wikipedia.org/wiki/Locally_connected_space)

But this difference only holds when looking at a single point, and when either property holds at every point, the two notions become identical. In pi-base, we only care about the property at every point.

[GeoffreySangston]

@prabau @StevenClontz Anyone reading what I wrote should note that there's a few subtle mistakes I'm currently fixing (Edit: Okay well I'm done editing it for now).

I do want to mention this:

(A) $X$ is ('Borsuk') locally contractible at $x$ if each neighborhood $U$ of $x$ contains a neighborhood $V$ which is deformable in $U$ to a point $x' \in U$.

(B) $X$ is ('open Borsuk') locally contractible' at $x$ if each neighborhood $U$ of $x$ contains an open neighborhood $V$ which is deformable in $U$ to a point $x' \in U$.

Am I right that (A) <=> (B)?: It's clear that (B) implies (A). Suppose (A). Then there exists a homotopy $V \times [0, 1] \to U$. There is an open neighborhood $V' \subset V$ of $x$. Restrict the homotopy to $V' \times [0, 1] \to U$. This means (B) holds.

(Just trying to be ultra careful since there seems to be a lot of room for error here. Another potential difficulty when comparing results is a lot of older sources assume Hausdorff (Borsuk does this) or $T_0$ in the definition of a space.)

[GeoffreySangston]

Thanks. I have not thought much about any of that, just giving some food for thought about the different possibilities one could use. And possibly (or not) introduce multiple variants to illustrate the differences. It looks like it's not a small project.

Having finished the edits, it does seem complicated. I would like to ask mathoverflow about the distinctions eventually, if I find time to make my own effort first. (Well I may as well start that now..)

An immediate goal I thought made sense, hence the reason for this, was to get something stronger than locally simply connected and semilocally simply connected (which I'll have to look into ensure there aren't additional quagmires there) into pi-base.

• Okay well Encyclopedia of General Topology's version of locally simply connected doesn't (obviously?) appear to be the same as 'locally' simply connected. So further quagmires await. It appears to also be taken from Borsuk though, but I'll say no more here in order not to distract us further.

[GeoffreySangston]

Another update. Now I believe properties 1, 1', and 1'' are equivalent, which recombines most of what was in the original 1 anyways. See my MSE post (which is long but basic).

Also, I want to quote Paul Frost and the anonymous Moishe Kohan from ANR is locally contractible(https://math.stackexchange.com/a/3926424/444923), whose opinions I trust:

First Paul Frost:

It seems that there are two rivalling definitions of "locally contractible":

1. $Y$ is locally contractible if each $y_0 \in Y$ has arbitrarily small (open) contractible neigborhoods.

2. For each $y_0 \in Y$ and each open neigborhood $U$ of $y_0$ in $Y$ there exists an open neighborhood $V$ of $y_0$ in $Y$ which is contained in $U$ such that the inclusion $V \hookrightarrow U$ is null-homotopic.

In my opinion 2. is the standard definition. Clearly 1. implies 2., but I doubt that the converse is true. ...
...
The concept of local contractibility was introduced by K. Borsuk in the nineteen-thirties. See

Borsuk, K. "Über eine Klasse von lokal zusammenhängenden Räumen." Fund. Math 19 (1932): 220-242.

Then Moishe Kohan:

Definition 2 should be regarded as the standard one, it is the one introduced by Borsuk and used to prove that each ANR is locally contractible.

Definition 1 is not equivalent to Definition 2, see two examples here: These are ANRs (even ARs) which fail Definition 1.

I think this really is a strong case that the most important definition is Borsuk's.