Comb space

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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 ${\mathbb{R}}^2$ 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 integers.

Topological space properties

Properties it does satisfy

• Closed sub-Euclidean space
• Metrizable space
• Contractible space
• SDR-contractible space, viz it admits the origin as a strong deformation retract, via the homotopy that first collapses all the vertical line segments to the $x$-axis, and then collapses the entire $x$-axis to the origin.

Properties it does not satisfy

• Suddenly contractible space: There is no contracting homotopy that is a homeomorphism for all $t<1$
• Everywhere SDR-contractible space: In particular, the point $(0,1)$ is not a strong deformation retract
• Locally path-connected space: The neighbourhood of $(0,1)$ has infinitely many path components.

Related spaces

• Double-comb space which is a weakly contractible space that is not contractible
• Broom space