A space $X$ is called locally contractible it it has a basis of neighbourhoods which are themselves contractible spaces. CW complexes and manifolds are locally contractible. On the other hand, the path fibration $PX \to X$ space of based paths with evaluation at the endpoint as projection) admits local sections iff $X$ is $\infty$-well-connected (or locally relatively contractible, or semi-locally contractible), that is, has a basis of neighbourhoods $N$ such that the inclusion maps $N\hookrightarrow X$ are null homotopic. Another use of this concept is by Dold, when he proves a Dold fibration (a map with the Weak Covering Homotopy Property) over an $\infty$-well-connected space is locally homotopy trivial.
What, then, is an example of a space which is $\infty$-well-connected but not locally contractible?
Edit: Note that the 1-dimensional version of this is a space that is semilocally 1-connected (or 1-well-connected, in my revisionist terminology), but not locally 1-connected.
