I'm looking for an example of a metric space $X$ that is not contractible but satisfies the following "uniform contractability" condition: for every $r>0$ there exists $s\geq r$ such that for any $x\in X$, the ball $B_r(x)$ is contractible inside the larger ball $B_s(x)$.
Clearly, if $X$ has finite diameter then contractibility is equivalent to uniform contractibility. Also, if $X$ has the homotopy type of a CW-complex, then by Whitehead's theorem, $X$ being uniformly contractible implies that it is in fact contractible. So one needs to look at metric spaces with infinite diameter that are not homotopy equivalent to a CW-complex.
So far I've seen two main examples I've seen (which don't quite satisfy the criteria) are:
- The long line, which is not homotopy equivalent to a CW-complex but is also not metrizable;
- The Warsaw circle (also known as the quasi-circle), which is a compact metric space not homotopy equivalent to a CW-complex.
Question 1: What is an example of a metric space $X$ with infinite diameter that is not homotopy equivalent to a CW-complex?
Question 2: Are there examples of such spaces that are uniformly contractible but not contractible?