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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.

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up vote 6 down vote accepted

The same counterexample as for semilocally 1-connected works: namely, you can take the cone on the Hawaiian earring space. The space itself is contractible, but no sufficiently small neighborhoods of the "bad" point at the base of the cone are 1-connected (hence not contractible).

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Ah, it's so obvious! I was thinking of something that had homotopical obstructions in all dimensions, but this is short and sweet! – David Roberts May 27 '10 at 4:56

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