The union of growing circles is not homeomorphic to wedge sum of circles
Let $X$ be the subspace of $\mathbb{R}^2$ that is the union of the circles $C_n$ of radius $n$ and center $(n,0)$ for $n \in \mathbb{N}$. Show that $X$ and $\bigvee_\infty S^1$ are homotopy equivalent, but not homeomorphic.
It is unclear to me why the "obvious" maps between the two spaces do not give a homeomorphism. It seems like the difference would have to be a result of the behavior of the map near the origin. Could I get a hint?
Solution 1:
HINT: $X$ is first countable. The common point in the wedge does not have a countable local base.