An example of a space which is locally relatively contractible but not contractible? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T11:17:03Z http://mathoverflow.net/feeds/question/26104 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/26104/an-example-of-a-space-which-is-locally-relatively-contractible-but-not-contractib An example of a space which is locally relatively contractible but not contractible? David Roberts 2010-05-27T04:18:04Z 2010-05-27T04:47:02Z <p>A space $X$ is called <em>locally contractible</em> 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 <em>$\infty$-well-connected</em> (or <em>locally relatively contractible</em>, or <em>semi-locally contractible</em>), 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.</p> <p>What, then, is an example of a space which is $\infty$-well-connected but not locally contractible?</p> <hr> <p>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.</p> http://mathoverflow.net/questions/26104/an-example-of-a-space-which-is-locally-relatively-contractible-but-not-contractib/26107#26107 Answer by Tyler Lawson for An example of a space which is locally relatively contractible but not contractible? Tyler Lawson 2010-05-27T04:47:02Z 2010-05-27T04:47:02Z <p>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).</p>