# Comb space

From Topospaces

This article is about a particular topological space (uniquely determined up to homeomorphism)|View a complete list of particular topological spaces

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

## Contents

## Definition

The comb space is defined as the following subset of with the subspace topology: It is the union of , , and all line segments of the form where varies over the positive 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 -axis, and then collapses the entire -axis to the origin.

### Properties it does not satisfy

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

## Related spaces

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