Difference between revisions of "Connected topological group"

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(New page: {{topological group property}} ==Definition== A topological group is termed '''connected''' if it satisfies the following equivalent conditions: * It is connected as a topological s...)
 
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==Definition==
 
==Definition==
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===Symbol-free definition===
  
 
A [[topological group]] is termed '''connected''' if it satisfies the following equivalent conditions:
 
A [[topological group]] is termed '''connected''' if it satisfies the following equivalent conditions:
  
* It is connected as a topological space.
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# It is connected as a topological space.
* It has no proper [[open subgroup]]
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# It has no proper [[open subgroup]]
* The connected component of the identity element equals the whole group.
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# The connected component of the identity element equals the whole group.
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===Equivalence of definitions===
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Definitions (1) and (3) are clearly equivalent. For proof of (1) implies (2), use [[connected implies no proper open subgroup]]. The reverse implication is somewhat trickier.

Revision as of 19:41, 22 June 2008

This article defines a property that can be evaluated for a topological group (usually, a T0 topological group)
View a complete list of such properties

Definition

Symbol-free definition

A topological group is termed connected if it satisfies the following equivalent conditions:

  1. It is connected as a topological space.
  2. It has no proper open subgroup
  3. The connected component of the identity element equals the whole group.

Equivalence of definitions

Definitions (1) and (3) are clearly equivalent. For proof of (1) implies (2), use connected implies no proper open subgroup. The reverse implication is somewhat trickier.