I am looking for an example of a well-pointed space in which no (sufficiently small) neighbourhood of the base-point is contractible. As usual, a well-pointed space is a pointed space in which the inclusion of the base-point is a Hurewicz cofibration.


In the reverse direction, I am looking for conditions under which a well-pointed space has a basis of contractible neighbourhoods around the base-point.



Well-pointed space which is not locally contractible
My initial answer was fully ignorant of the literature on the subject; I'm now recalling that it does exist. In 1934, Borsuk and Mazurkiewicz constructed a $2$-dimensional AR (=contractible, locally contractible $2$-dimensional compactum) $X$ that is not a countable union of smaller ARs (Sur les rétractes absolus indécomposables, C.R.. Acad. Sci. Paris 199 (1934), 110-112; see Borsuk's "Theory of retracts", Section VI.4). The proof actually shows that $X$ is not a countable union of contractible proper subsets (open or not). Thus $X$ contains points that do not have a basis of contractible neighborhoods. 


A simplified version $Y$ of $X$ (constructed below), which is an AR failing to have a basis of contractible neighborhoods at a certain point $\infty$, is also forward tame at $\infty$ and so answers Ricardo's [revised question](https://lists.lehigh.edu/pipermail/algtop-l/2012q1/001331.html) at the Algebraic Topology Discussion List. $Y$ looks somewhat like a $2$-skeleton of $W^+$, the example in my previous answer, except that $W^+$ is not forward tame. A google search for "Mazurkiewicz singularity" (this is how Borsuk called the kind of ANRs that do not have arbitrarily fine countable covers by ARs) returns a paper by [Armentrout](http://matwbn.icm.edu.pl/ksiazki/fm/fm69/fm69110.pdf) which has an example somewhat similar to $W^+$ (also a decomposition space and a manifold factor) though more laborious.


Note also a positive result of [Begle](http://projecteuclid.org/euclid.bams/1183511682) proved using Zorn's lemma.




---


Here's a construction of $X$ (which I hope is easier to read than Borsuk's account) and also of $Y$. 


Let $Q=D^2/\{a,b\}$, the quotient of the $2$-disk where two distinct interior points $a,b$ are identified with each other. Let $\partial Q$ be the image of $\partial D^2$ in $Q$, and let $C$ be the image in $Q$ of a path joining $a$ and $b$ in the interior of $D^2$. Let $(Q\_1,\partial Q\_1)=(D^2,\partial D^2)$, and form $(Q\_k,\partial Q\_k)$ by attaching $Q\_{k-1}$ to $(Q,\partial Q)$ along a homeomorphism $\partial Q\_{k-1}\to C$. Then $Q\_k$ is contractible, but no proper subset of $Q\_k$ containing $\partial Q\_k$ is contractible (or even acyclic).


Now $Y$ is just the one point compactification $(Q\_\infty)^+$ of the direct limit $Q\_\infty$ of the chain of inclusions $Q\_1\subset Q\_2\subset\dots$.


To see that $Y$ is locally contractible, observe that $D^2\times [0,1]$ collapses onto a copy of $Q\_2$ such that $\partial Q\_2=\partial D^2\times\{1\}$ and $Q\_1=D^2\times\{0\}$. It follows that $D^2\times [0,\infty)$ admits a proper deformation retraction onto a copy of $Q\_\infty$. Thus $Q\_\infty$ is proper homotopy equivalent to $[0,\infty)$. Then $Q\_\infty$ is forward tame in the sense of Quinn (see Definition 7.1, p.75 and Proposition 9.6, p.99 in [Hughes-Ranicki](http://www.maths.ed.ac.uk/~aar/books/ends.pdf)), and therefore its one-point compactification is locally contractible.


For the construction of $X$ we will need $k$ copies $C\_1^{Q\_k},\dots,C\_k^{Q\_k}$ of $S^1$ embedded in $Q\_k$, namely $C\_1^{Q\_k}=\partial Q\_k$, and $C\_{i+1}^{Q\_k}=C\_i^{Q\_{k-1}}$.
Now $X$ is formed by attaching increasingly smaller copies of $Q\_1,Q\_8,Q\_{64},...$ to the [Sierpinsky carpet](http://en.wikipedia.org/wiki/Sierpinski_carpet) so that for each $k$ of the form $8^m$, the $k$ circles $C\_1^{Q\_k},\dots,C\_k^{Q\_k}$ in $Q\_k$ are attached to the boundaries of the disks removed at the $m$th stage of the construction of the Sierpinski carpet, and additionally $(k-1)$ arcs connecting $C\_i^{Q\_k}$ with $C\_{i+1}^{Q\_{k+1}}$ are identified with analogous arcs in the Sierpinsky carpet. This can, and must be done so that every neighborhood of every point of the Sierpinsky carpet contains $C\_1^{Q\_k}=\partial Q\_k$ for some $k$ (of the form $8^m$).


