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Concept
Comb space
Summary
In mathematics, particularly topology, a comb space is a particular subspace of 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.
Consider with its standard topology and let K be the set . The set C defined by:
considered as a subspace of 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.
The comb space and the deleted comb space have some interesting topological properties mostly related to the notion of connectedness.
- The comb space, C, is path connected and contractible, but not locally contractible, locally path connected, or locally connected.
- The deleted comb space, D, is connected: Let E be the comb space without . E is also path connected and the closure of E is the comb space. As E D the closure of E, where E is connected, the deleted comb space is also connected.
- 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 f : [0, 1] → D be this path. We shall prove that −1{p} is both open and closed in [0, 1] contradicting the connectedness of this set. Clearly we have f −1{p} is closed in [0, 1] by the continuity of f. To prove that f −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 f −1{p}. Clearly, f(x) = p. Then since f −1(V) is open, there is a basis element U containing x such that f(U) is a subset of V. We assert that f(U) = {p} which will mean that U is an open subset of f −1{p} containing x. Since x was arbitrary, f −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, f(U) is connected. Suppose f(U) contains a point s other than p.
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