Double comb space

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 double comb space is a subset of $\mathbb {R} ^{2}$ obtained by taking the standard comb space, and attaching another comb space to it at $(0,1)$, where the new comb space is obtained by reflecting the previous one about the point $(0,1)$.