Electron degeneracy pressure: Difference between revisions

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{{morefootnotes|date=September 2011}}
In [[topology]], a '''clopen set''' (a [[portmanteau]] of '''closed-open set''') in a [[topological space]] is a set which is both [[open set|open]] and [[closed set|closed]].  That this is possible may seem counter-intuitive, as the common meanings of ''open'' and ''closed'' are antonyms. But their mathematical definitions are not [[mutually exclusive]].  A set is closed if its [[Complement_(set_theory)|complement]] is open, which leaves the possibility of an open set whose complement is itself also open, making both sets both open ''and'' closed, and therefore clopen.
 
== Examples ==
 
In any topological space ''X'', the [[empty set]] and the whole space ''X'' are both clopen.<ref>{{cite book|author=Bartle, Robert G.|coauthors=Sherbert, Donald R.|title=Introduction to Real Analysis|edition=2nd|publisher=John Wiley & Sons, Inc.|origyear=1982|year=1992|pages=348 (regarding the real numbers and the empty set in R)}}</ref><ref>{{cite book|author=Hocking, John G.|coauthors=Young, Gail S.|title=Topology|publisher=Dover Publications, Inc.|location=NY|year=1961|pages=5 and 6 (regarding topological spaces)}}
</ref>
 
Now consider the space ''X'' which consists of the union of the two open [[interval (mathematics)|interval]]s (0,1) and (2,3) of '''R'''. The topology on ''X'' is inherited as the [[topological subspace|subspace topology]] from the ordinary topology on the [[real line]] '''R'''. In ''X'', the set (0,1) is clopen, as is the set (2,3). This is a quite typical example: whenever a space is made up of a finite number of disjoint [[connected space|connected component]]s in this way, the components will be clopen.
 
As a less trivial example, consider the space '''Q''' of all [[rational number]]s with their ordinary topology, and the set ''A'' of all positive rational numbers whose square is bigger than 2. Using the fact that <math>\sqrt 2</math> is not in '''Q''', one can show quite easily that ''A'' is a clopen subset of '''Q'''. (Note also that ''A'' is ''not'' a clopen subset of the real line '''R'''; it is neither open nor closed in '''R'''.)
 
==Properties==
 
* A topological space ''X'' is [[connected space|connected]] if and only if the only clopen sets are the empty set and ''X''.
* A set is clopen if and only if its [[boundary (topology)|boundary]] is empty.
* Any clopen set is a union of (possibly infinitely many) [[Connected space|connected components]].
* If all connected components of ''X'' are open (for instance, if ''X'' has only finitely many components, or if ''X'' is [[locally connected]]), then a set is clopen in ''X'' if and only if it is a union of connected components.
* A topological space ''X'' is [[discrete space|discrete]] if and only if all of its subsets are clopen.
* Using the [[union (set theory)|union]] and [[intersection (set theory)|intersection]] as operations, the clopen subsets of a given topological space ''X'' form a [[Boolean algebra (structure)|Boolean algebra]]. ''Every'' Boolean algebra can be obtained in this way from a suitable topological space: see [[Stone's representation theorem for Boolean algebras]].
 
==Notes==
{{reflist}}
 
==References==
*{{cite web|url=http://uob-community.ballarat.edu.au/~smorris/topology.htm|title=Topology Without Tears|author=Sidney A. Morris|archiveurl=http://web.archive.org/web/20130419134743/http://uob-community.ballarat.edu.au/~smorris/topology.htm|archivedate=19 April 2013}}
 
[[Category:General topology]]

Revision as of 22:35, 17 January 2014

Template:Morefootnotes In topology, a clopen set (a portmanteau of closed-open set) in a topological space is a set which is both open and closed. That this is possible may seem counter-intuitive, as the common meanings of open and closed are antonyms. But their mathematical definitions are not mutually exclusive. A set is closed if its complement is open, which leaves the possibility of an open set whose complement is itself also open, making both sets both open and closed, and therefore clopen.

Examples

In any topological space X, the empty set and the whole space X are both clopen.[1][2]

Now consider the space X which consists of the union of the two open intervals (0,1) and (2,3) of R. The topology on X is inherited as the subspace topology from the ordinary topology on the real line R. In X, the set (0,1) is clopen, as is the set (2,3). This is a quite typical example: whenever a space is made up of a finite number of disjoint connected components in this way, the components will be clopen.

As a less trivial example, consider the space Q of all rational numbers with their ordinary topology, and the set A of all positive rational numbers whose square is bigger than 2. Using the fact that 2 is not in Q, one can show quite easily that A is a clopen subset of Q. (Note also that A is not a clopen subset of the real line R; it is neither open nor closed in R.)

Properties

  • A topological space X is connected if and only if the only clopen sets are the empty set and X.
  • A set is clopen if and only if its boundary is empty.
  • Any clopen set is a union of (possibly infinitely many) connected components.
  • If all connected components of X are open (for instance, if X has only finitely many components, or if X is locally connected), then a set is clopen in X if and only if it is a union of connected components.
  • A topological space X is discrete if and only if all of its subsets are clopen.
  • Using the union and intersection as operations, the clopen subsets of a given topological space X form a Boolean algebra. Every Boolean algebra can be obtained in this way from a suitable topological space: see Stone's representation theorem for Boolean algebras.

Notes

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References

  1. ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

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  2. ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
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