Comb space

In mathematics, particularly topology, a comb space is a particular subspace of formula_1 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 formula_1 with its standard topology and let "K" be the set formula_3. The set "C" defined by:
considered as a subspace of formula_1 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 formula_7 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 is an example of a path connected space which is not locally path 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 deformation retract onto a point for every choice of basepoint.