Comb space

In mathematics, particularly topology, a comb space is a particular subspace of ${\displaystyle \mathbb {R} ^{2}}$ 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.

Topologist's comb

The intricated double comb for r=3/4.

Formal definition

Consider ${\displaystyle \mathbb {R} ^{2}}$ with its standard topology and let K be the set ${\displaystyle \{1/n~|~n\in \mathbb {N} \}}$. The set C defined by:

${\displaystyle (\{0\}\times [0,1])\cup (K\times [0,1])\cup ([0,1]\times \{0\})}$

considered as a subspace of ${\displaystyle \mathbb {R} ^{2}}$ equipped with the subspace topology is known as the comb space. The deleted comb space, D, is defined by:

${\displaystyle (\{0\}\times \{0,1\})\cup (K\times [0,1])\cup ([0,1]\times \{0\})}$.

This is the comb space with the line segment ${\displaystyle \{0\}\times (0,1)}$ 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:

Let E be the comb space without ${\displaystyle \{0\}\times (0,1]}$. E is also path connected and the closure of E is the comb space. As E ${\displaystyle \subset }$ D ${\displaystyle \subset }$ the closure of E, where E is connected, the deleted comb space is also connected.

3. The deleted comb space is not path connected since there is no path from (0,1) to (0,0):

Suppose there is a path from p = (0, 1) to the point (0, 0) in D. Let ƒ : [0, 1] → D be this path. We shall prove that ƒ ⁻¹{p} is both open and closed in [0, 1] contradicting the connectedness of this set. Clearly we have ƒ ⁻¹{p} is closed in [0, 1] by the continuity of ƒ. To prove that ƒ ⁻¹{p} is open, we proceed as follows: Choose a neighbourhood V (open in R²) about p that doesn’t intersect the x–axis. Suppose x is an arbitrary point in ƒ ⁻¹{p}. Clearly, f(x) = p. Then since f ⁻¹(V) is open, there is a basis element U containing x such that ƒ(U) is a subset of V. We assert that ƒ(U) = {p} which will mean that U is an open subset of ƒ ⁻¹{p} containing x. Since x was arbitrary, ƒ ⁻¹{p} will then be open. We know that U is connected since it is a basis element for the order topology on [0, 1]. Therefore, ƒ(U) is connected. Suppose ƒ(U) contains a point s other than p. Then s = (1/n, z) must belong to D. Choose r such that 1/(n + 1) < r < 1/n. Since ƒ(U) does not intersect the x-axis, the sets A = (−∞, r) × ${\displaystyle \mathbb {R} }$ and B = (r, +∞) × ${\displaystyle \mathbb {R} }$ will form a separation on f(U); contradicting the connectedness of f(U). Therefore, f ⁻¹{p} is both open and closed in [0, 1]. This is a contradiction.

4. The comb space is homotopic to a point but does not admit a deformation retract onto a point for every choice of basepoint.

See also

• Connected space
• Hedgehog space
• Infinite broom
• List of topologies
• Locally connected space
• Order topology
• Topologist's sine curve

References

• James Munkres (1999). Topology (2nd ed.). Prentice Hall. ISBN 0-13-181629-2.

• Kiyosi Itô (ed.). "Connectedness". Encyclopedic Dictionary of Mathematics. Mathematical Society of Japan. Cite journal requires |journal= (help)