Finding homotopy equivalence

Question

This is part of a problem from Hatcher: Show that the space in $\mathbb R^2$ which is the union (for $n \in \mathbb N$) of circles $C_n$, where $C_n$ is the circle centered at $(n,0)$ with radius $n$ is not homoemorphic to the wedge sum of infinite circles, but they are homotopy equivalent.

I was able to prove that these spaces are not homeomorphic by considering how open sets at $0$ for the first case would differ from the open sets around wedge point.

But I have no idea how to prove (in this case and in general too) how to prove that two spaces are homotopically equivalent if the spaces under consideration are not simple CW complexes.

Thanks!

Answer 1

The obvious "identity" from the wedge to the subset of R^2 is continuous but not a homeomorphism (there is a problem at the point O). Then you must find an "inverse" map g such that gof is not the identity but "homotopic" to the identity... Where you should send O in order not to have trouble?