# Contractible and Delta-generated implies strong deformation retract to a point?

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?

• Maybe general topology tag? – Fernando Muro May 7 '15 at 20:17
• 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. – HenrikRüping May 8 '15 at 4:28
• Of course an example of the latter kind would be ideal, but I would also be interested in an example of the former kind. – Chris Schommer-Pries May 8 '15 at 7:34
• 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. – Gabriel C. Drummond-Cole May 8 '15 at 9:06
• I haven't checked but perhaps Gabriel's example generalizes to something like $\{ (x, y) \in [0,\infty)^2 \mid y \le x\} / ((x, y) \sim (x, y')\text{ if } y, y' < x)$. – 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
(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.