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? 

 A: 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. 
