[ALGTOP-L] Question: non-locally contractible well-pointed space

[Ricardo Andrade]

Hi all,

I don't like spamming the list with questions. Nevertheless, I hope the 
people here might be better equipped to answer my question.

I recently posted a question on mathoverflow

http://mathoverflow.net/questions/87387/well-pointed-space-which-is-not-locally-contractible

It asks specifically for an example of a well-pointed space in which no 
(sufficiently small) neighbourhood of the base-point is contractible.

I did get a rather unexpected answer which is indeed very interesting, but 
fails to do what I wanted. It seems to me that such a counter-example 
would be a natural thing to look for. So I come to this list to see if 
someone knows of another example of a well-pointed space for which there 
is no basis of contractible neighbourhoods around the base-point.

[Real goal: I actually want the inclusion of the base-point to be tame in 
the sense of Quinn's "Homotopically stratified sets". This means that in 
the NDR-pair characterization of cofibrations, the homotopy H which 
appears also verifies: H(x,t)=* (* is the base-point) implies x=* or t=1.]

Thanks in advance,
Ricardo