If a CW-complex is contractible, then it strongly deformation retracts onto the inclusion of a point.

However for general spaces it is well-known that just because a space is contractible, it does not follow that it strongly deformation retracts to a point. Here I am using the Wikipedia terminology. An example can be found in Hatcher's "Algebraic Topology" (pg18) (see this question). But of course this example is a bit pathological.

I am wondering just how bad these examples have to be?

A nice class of spaces which is much more general that CW-complexes, but discards many pathological examples is the class of $\Delta$-generated spaces. For example the $\Delta$-generation of the rationals or the cantor sets are just discrete sets. Such spaces disappear in the $\Delta$-generated category, and so does Hatcher's counter example.

Is there an example of a contractible $\Delta$-generated space which fails to deformation retract onto a point?

  • $\begingroup$ Maybe general topology tag? $\endgroup$ – Fernando Muro May 7 '15 at 20:17
  • $\begingroup$ Just to be clear? Are you asking for a contractible space and a point in it such that there is no deformation retraction or for a space that does not deformation retract to any point in it? I assume its the latter. $\endgroup$ – HenrikRüping May 8 '15 at 4:28
  • $\begingroup$ Of course an example of the latter kind would be ideal, but I would also be interested in an example of the former kind. $\endgroup$ – Chris Schommer-Pries May 8 '15 at 7:34
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    $\begingroup$ The interval with two endpoints $\left([0,1]\times\{a,b\}\right)/ \left((x,a)\sim (x,b)\text{ if }x\ne 0\right)$ is an example of the former kind. $\endgroup$ – Gabriel C. Drummond-Cole May 8 '15 at 9:06
  • $\begingroup$ I haven't checked but perhaps Gabriel's example generalizes to something like $\{ (x, y) \in [0,\infty)^2 \mid y \le x\} / ((x, y) \sim (x, y')\text{ if } y, y' < x)$. $\endgroup$ – Karol Szumiło May 8 '15 at 14:54

I believe the "complete feather", a non-hausdorff $1$-manifold due to Haefliger and Reeb, is such an example. It is a simple generalization of Gabriel C. Drummond-Cole example of the interval with two endpoints. Actually, we discuss it in

Mathieu Baillif, Alexandre Gabard, Manifolds: Hausdorffness versus homogeneity, Proceedings of the American Mathematical Society 136 no 3 (2008) pp 1105–1111, doi:10.1090/S0002-9939-07-09100-9, arXiv:math/0609098.

(sorry for the self-promotion). This space is $\Delta$-generated, if I am not mistaken.

Say that a space $X$ is locally strongly contractible if each point $x\in X$ has a neighborhood which strongly deformation retracts to $x$. (The retraction needs to be defined only in the neighborhood, not the whole space.) David Gauld proved long ago the following theorem: If a space $X$ is locally strongly contractible, contractible to a point $p$, and completely regular at $p$, then $X$ strongly deformation retracts to $p$. The complete feather shows that you cannot drop entirely the "completely regular" assumption.

  • $\begingroup$ The complete feather is a quotient of a Delta-generated space (a disjoint union of real lines) by an equivalence relation so it's Delta-generated. $\endgroup$ – Gabriel C. Drummond-Cole May 10 '15 at 18:35

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