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](http://en.wikipedia.org/wiki/Deformation_retract). An example can be found in Hatcher's "Algebraic Topology" (pg18) (see [this question](https://math.stackexchange.com/questions/150751/contractible-vs-deformation-retract-to-a-point)). 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](http://ncatlab.org/nlab/show/Delta-generated+space). 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. 



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



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> 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](https://doi.org/10.1090/S0002-9939-07-09100-9), arXiv:[math/0609098](https://arxiv.org/abs/math/0609098).
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(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. 


