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In [[mathematics]], particularly [[topology]], a '''comb space''' is a [[subspace topology|subspace]] of <math>\R^2</math> 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.
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[[File:Comb.svg|right|thumb|Topologist's comb]]
[[File:Double comb.svg|right|thumb|The intricated double comb for r=3/4.]]
==Formal definition==
 
Consider <math>\R^2</math> with its [[standard topology]] and let ''K'' be the [[Set (mathematics)|set]] <math>\{1/n | n \in \mathbb N\}</math>. The set ''C'' defined by:
 
:<math>(\{0\} \times [0,1] ) \cup (K \times [0,1]) \cup ([0,1] \times \{0\})</math>
 
considered as a subspace of <math>\R^2</math> equipped with the [[subspace topology]] is known as the comb space. The deleted comb space, D, is defined by:
 
:<math>(\{0\} \times \{0,1\} ) \cup (K \times [0,1]) \cup ([0,1] \times \{0\})  </math>.
 
This is the comb space with the line segment <math>\{0\} \times (0,1)</math> deleted.
 
==Topological properties==
 
The comb space and the deleted comb space have some interesting topological properties mostly related to the notion of [[locally connected space|connectedness]].  
 
1. The comb space is an example of a path connected space which is not [[Locally connected space|locally path connected]]; see the page on [[locally connected space]]s.
 
2. The deleted comb space, D, is connected. To show this, let E be the comb space without <math>\{0\} \times (0,1] </math>.  E is also path connected and the [[Closure (topology)|closure]] of E is the comb space. As E <math>\subset</math> D <math>\subset</math> 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 (topology)|path]] from (0,1) to (0,0):
 
Suppose there is a path from ''p'' = (0,&nbsp;1) to a point ''q'' in ''D''. Let ƒ:[0,&nbsp;1]&nbsp;→&nbsp;''D'' be this path. We shall prove that ƒ<sup>&nbsp;&minus;1</sup>{''p''} is both [[Open set|open]] and [[Closed set|closed]] in [0,&nbsp;1] contradicting the [[connected space|connectedness]] of this set. Clearly we have ƒ<sup>&nbsp;&minus;1</sup>{''p''} is closed in [0,&nbsp;1] by the [[Continuous function|continuity]] of ƒ. To prove that ƒ<sup>&nbsp;&minus;1</sup>{''p''} is open, we proceed as follows: Choose a [[Neighbourhood (mathematics)|neighbourhood]] V (open in '''R'''<sup>2</sup>) about ''p'' that doesn’t intersect the ''x''–axis. Then there is a [[basis (topology)|basis]] element ''U'' containing ƒ<sup>&nbsp;&minus;1</sup>{''p''} such that ƒ(''U'') is a subset of ''V''. We know that U is connected since it is a basis element for the [[order topology]] on [''a'',&nbsp;''b'']. Therefore, ƒ(''U'') is connected. We assert that ƒ(''U'') = {''p''} so that ƒ<sup>&nbsp;&minus;1</sup>{''p''} is open. Suppose ƒ(''U'') contains a point (''z'',&nbsp;1/''n'') other than ''p''. Then (''z'',&nbsp;1/''n'') must belong to ''D''. Choose ''r'' such that 1/(''n''&nbsp;&minus;&nbsp;1) > ''r'' > 1/''n''. Since ƒ(''U'') doesn’t intersect the ''x''-axis, the sets:
 
: ''A'' = (&minus;&infin;, ''r'') &times; ''R''
 
: ''B'' = (''r'', +&infin;) &times; ''R''
 
will form a [[separated sets|separation]] on ''f''(''U''); contradicting the connectedness of ''f''(''U''). Therefore, ''f''<sup>&nbsp;&minus;1</sup>{''p''} is both open and closed in [0,&nbsp;1]. This is a contradiction.
 
==See also==
* [[Locally connected space]]
* [[Connected space]]
* [[Topologist's sine curve]]
* [[Infinite broom]]
* [[Order topology]]
 
==References==
* {{cite book
| author = [[James Munkres]]
| year = 1999
| title = Topology
| edition = 2nd
| publisher = [[Prentice Hall]]
| isbn = 0-13-181629-2
}}
 
*{{Cite journal|title=Connectedness|series=Encyclopedic Dictionary of Mathematics|publisher=Mathematical Society of Japan|editor=Kiyosi Ito}}
 
[[Category:Topological spaces]]
[[Category:Trees (topology)]]

Revision as of 12:44, 4 March 2014

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