Comb space

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.



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