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I'm working on formalizing locally contractible spaces in Mathlib (the mathematics library for the Lean theorem prover), and I've encountered conflicting terminology in the literature regarding local contractibility. I'd like to clarify the standard names for these notions.

Three related properties:

Consider the following three properties for a topological space X (listed from weakest to strongest):

  1. Null-homotopic inclusions: For every point x ∈ X and every neighborhood U of x, there exists a neighborhood V of x with V ⊆ U such that the inclusion map V ↪ U is null-homotopic.
  2. Contractible neighborhood basis: Every point has a neighborhood basis consisting of (not necessarily open) contractible sets.
  3. Open contractible neighborhood basis: Every point has a neighborhood basis consisting of open contractible sets.

The terminology question:

What are the standard names for these three properties in the topology literature?

From what I've gathered:

  • Property (1) is sometimes called "locally contractible" (LC)
  • Property (2) is sometimes called "strongly locally contractible" (SLC)
  • The https://ncatlab.org/nlab/show/locally+contractible+space suggests that property (1) might be called "semi-locally contractible" in some sources
  • Some authors may use "locally contractible" to mean property (2) rather than (1)

Complications:

  • I've been told that a counterexample by Borsuk and Mazurkiewicz (1934) shows that (1) does not imply (2), though I haven't verified the construction myself
  • It's unclear whether (2) and (3) are equivalent (can one always shrink a contractible neighborhood to an open contractible neighborhood?) (This was asked about at Definitions of locally contractible spaces, but without any answer.)

What I'm looking for:

  1. What are the most widely accepted/standard names for properties (1), (2), and (3)?
  2. Are there definitive references (textbooks, papers) that establish standard terminology?
  3. If there are regional or subfield differences in terminology, what should be considered the "default" for a general mathematics library?

Any guidance, particularly with references to standard topology texts (Hatcher, May, Spanier, etc.), would be greatly appreciated.

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  • $\begingroup$ I think (1) as you wrote it would be confusing to label as "semi-locally contractible". I would think semi-locally contractible should be analogous with semi-locally simply connected, where a nullhomotopy can move through all of X, and does not need to be able to be confined to any given U. So I would want to say that a cone on the hawaiian earring is semi-locally contractible. But this is just my thought, not based on any review of literature. $\endgroup$ Commented Nov 18 at 3:15
  • $\begingroup$ The nlab page you link seems to agree with me: semi-local contractibility includes any contractible space, and is about inclusions being nullhomotopic; it's not a local property like your (1). $\endgroup$ Commented Nov 18 at 3:26
  • $\begingroup$ There are some answers here: math.stackexchange.com/questions/1082601/… $\endgroup$ Commented Nov 18 at 4:36

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Here is what a few textbooks have to say:

From May's A Concise Course in Algebraic Topology, Chapter 4.1:

Observe that a graph is a locally contractible space: any neighborhood of any point contains a contractible neighborhood of that point. Therefore a connected graph has all possible covers.

  • May is the odd one out in this list, seems to only use local contractibility as a stronger-than-necessary property that implies the existence of a universal cover.

Theorem A.7 in Hatcher's Algebraic Topology:

A compact subspace $K$ of $\mathbb{R}^n$ is a retract of some neighborhood iff $K$ is locally contractible in the weak sense that for each $x \in K$ and each neighborhood $U$ of $x$ in $K$ there exists a neighborhood $V \subset U$ of $x$ such that the inclusion $V \hookrightarrow U$ is nullhomotopic.

Hatcher also has Proposition A.4, in which he avoids giving the stronger property a name:

Each point in a CW complex has arbitrarily small contractible open neighborhoods, so CW complexes are locally contractible.

Exercise 1C.6 in Spanier's Algebraic Topology book:

Prove that a binormal absolute neighborhood retract is locally contractible (that is, every neighborhood $U$ of a point $x$ contains a neighborhood $V$ of $x$ deformable to $x$ in $U$).

Remark following Corollary 8.31 in Rotman's An Introduction to Algebraic Topology:

A space $X$ is locally contractible if, for every $x \in X$, each neighborhood $U$ of $x$ contains an open neighborhood $V$ of $x$ that is contractible to $x$ in $U$; that is, there exists a continuous $F\colon V \times I \to U$ with $F(v, 0) = v$ and $F(v, 1) = x$ for all $v \in V$.

The start of Appendix E in Bredon's Topology and Geometry:

Moreover, since $\mathbb{R}^n$ is locally contractible, it is easy to see that $X$ is also locally contractible, meaning that any neighborhood $U$ of a point $x \in X$ contains a smaller neighborhood $V$ of $x$ such that there is a homotopy $F\colon V \times I \to U$ starting at the inclusion and ending at a constant map.

I would also refer to the answers at ANR is locally contractible. It seems that you are right that your (1) is the most standard definition of locally contractible.

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