Shrinking wedge of circles

Definition
The shrinking wedge of circles is the union of the circles in $$\R^2$$ having center $$(1/n,0)$$ and radius $$1/n$$, where $$n$$ is a positive integer. All these circles pass through the origin.

The space is connected, path-connected, and locally path-connected, but there is still a local convergence at the origin which cannot be captured by a CW-structure, so the shrinking wedge of circles is not a CW-space. In fact, its usual homology groups behave in a somewhat counterintuitive manner.

Related counterexamples

 * Shrinking wedge of spheres