Shrinking wedge of circles

This article describes a standard counterexample to some plausible but false implications. In other words, it lists a pathology that may be useful to keep in mind to avoid pitfalls in proofs
View other standard counterexamples in topology

Definition

The shrinking wedge of circles is the union of the circles in $\mathbb {R} ^{2}$ 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