In mathematics, particularly topology, a comb space is a particular subspace of \R^2 that resembles a comb. The comb space has properties that serve as a number of counterexamples. The topologist's sine curve has similar properties to the comb space. The deleted comb space is a variation on the comb space.

Formal definition

Consider \R^2 with its standard topology and let K be the set. The set C defined by: considered as a subspace of \R^2 equipped with the subspace topology is known as the comb space. The deleted comb space, D, is defined by: This is the comb space with the line segment deleted.

Topological properties

The comb space and the deleted comb space have some interesting topological properties mostly related to the notion of connectedness.

1. The comb space, C, is path connected and contractible, but not locally contractible, locally path connected, or locally connected.
2. The deleted comb space, D, is connected:
3. The deleted comb space is not path connected since there is no path from (0,1) to (0,0):
4. The comb space is homotopic to a point but does not admit a strong deformation retract onto a point for every choice of basepoint that lies in the segment