Harman Patil (Editor)

Comb space

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Comb space

In mathematics, particularly topology, a comb space is a subspace of R 2 that looks rather like a comb. The comb space has some rather interesting properties and provides interesting counterexamples. The topologist's sine curve has similar properties to the comb space. The deleted comb space is an important variation on the comb space.

Contents

Formal definition

Consider R 2 with its standard topology and let K be the set { 1 / n   |   n ∈ N ∖ { 0 } } . The set C defined by:

( { 0 } × [ 0 , 1 ] ) ∪ ( K × [ 0 , 1 ] ) ∪ ( [ 0 , 1 ] × { 0 } )

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:

( { 0 } × { 0 , 1 } ) ∪ ( K × [ 0 , 1 ] ) ∪ ( [ 0 , 1 ] × { 0 } ) .

This is the comb space with the line segment { 0 } × ( 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:

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

References

Comb space Wikipedia
(Text) CC BY-SA