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Recall that a space $X$ is contractible if there exists a homotopy $h:X\times [0,1]\to X$ such that $h$ is equal to the identity map on $X\times\{0\}$ and $h$ is constant on $X\times\{1\}$.

Please help me out since I'm stuck with this question, namely, why each ANR is locally contractible. It seems that I can prove that each AR is contractible this way: Consider $A = X\times \{0\} \cup X\times \{1\} \cup \{x_0\}\times [0,1] \subset X\times [0,1]$, where $x_0$ is any point from $X$. This set is closed in $X\times [0,1]$. Define $h: A\to X$ by $h|_{X\times \{0\}} = id_X$, $h|_{X\times \{1\}} = x_0$, $h|_{\{x_0\}\times [0,1]} = x_0$. This function is continuous, we can extend it (since being an AR is equivalent to being an AE) to the whole space $X\times [0,1]$. And this will be the required homotopy that squeezes $X$ into a point ($x_0$).

But similar approach seems not to work in case of an ANR. Because here I get a neighborhood $V$ of $A$ in $X\times [0,1]$ and an extension of $h$ onto this neighborhood. I can of course find an open $U\subset X$ such that $x_0\times [0,1] \subset U\times [0,1]\subset V$ (where $x_0$ is the point for which I want to find a contractible neighborhood) and consider the restriction of my extension onto $U\times [0,1]$, but the problem is that this map is not necessarily a map from $U\times [0,1]$ to $U$ (the range of my restricted extension can be larger than $U$).

Can anybody please give a hint on what's going on? (a hint would be even better than a full answer).

C_M
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2 Answers2

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It is surprising that your question has been unanswered for six years.

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. You have proved that ANRs are locally contractible in the sense of 2.

Many concrete examples of locally contractible spaces satisfy 1., e.g. manifolds and CW-complexes.

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.

Also see

Borsuk, Karol. Theory of retracts. Vol. 44. Państwowe Wydawn. Naukowe, 1967.

Paul Frost
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Just to clarify the terminological confusion, continuing with Paul Frost's notation:

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 am not sure who is responsible for renaming, but Hatcher in his "Algebraic Topology" refers to Definition 2 as "locally contractible in weak sense." But at least he spells out both definitions. In contrast, nlab here pretends that the standard definition (Definition 2) does not even exist. Ditto wikipedia here.

Moishe Kohan
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    It is interesting that in Hatcher's edition from 2002 one does not find a real definition of "locally contractible". He frequently uses the phrase, but what comes closest to a definition is "Each point in a CW complex has arbitrarily small contractible open neighborhoods, so CW complexes are locally contractible." (Proposition A.4). I think this doesn't allow a unique interpretation. However, in Theorem A.7 he defines "locally contractible in the weak sense", thus it is obvious that "locally contractible" must be something else. In the edition from 1999 this becomes clear: – Paul Frost Nov 28 '20 at 17:07
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    In Proposition A.3 he states "A CW complex is locally contractible, in the strong sense that each point has arbitrarily small contractible open neighborhoods." It therefore seems to me that a better approach would be to use the standard definition of "locally contractible" and to define "strongly locally contractible" in the obvious way. – Paul Frost Nov 28 '20 at 17:15
  • See also this question. It seems that for Steenrod a definition of "locally contractible" didn't deem necessary, probably because it had a standard interpretation. But as you pointed out, the present terminological status is confusing. – Paul Frost Nov 28 '20 at 17:30
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    Yes, I agree, it would be better to use "strongly locally contractible" for Definition 1 and keep "locally contractible" for Definition 2, at the very least, for historic reasons. And yes, at the time when Steenrod was writing, locally contractible was a standard definition (I think, it is even in the book by Alexandroff-Hopf). – Moishe Kohan Nov 28 '20 at 17:48