# 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 comb space is defined as the following subset of $\R^2$ with the subspace topology: It is the union of $[0,1] \times \{ 0 \}$, $\{ 0 \} \times [0,1]$, and all line segments of the form $\{ 1/n \} \times [0,1]$ where $n$ varies over the positive integers.