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...)
 
(Alternative definition for a locally connected topological group)
 
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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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# The [[connected component of identity|connected component of the identity element]] equals the whole group.
* 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 (2) are clearly equivalent.
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===Alternative definition for a locally connected topological group===
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For a [[locally connected topological group]], being connected is equivalent to having no proper open subgroup. See [[locally connected and no proper open subgroup implies connected]]
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In particular, this alternate definition applies to [[algebraic group]]s equipped with the Zariski topology, as well as to [[Lie group]]s. For more, see [[equivalence of definitions of connected algebraic group]] and [[equivalence of definitions of connected Lie group]].
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==Facts==
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* [[Connected implies no proper open subgroup]]: The converse holds for [[locally connected topological group]]s.

Latest revision as of 18:03, 13 January 2012

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. The connected component of the identity element equals the whole group.

Equivalence of definitions

Definitions (1) and (2) are clearly equivalent.

Alternative definition for a locally connected topological group

For a locally connected topological group, being connected is equivalent to having no proper open subgroup. See locally connected and no proper open subgroup implies connected

In particular, this alternate definition applies to algebraic groups equipped with the Zariski topology, as well as to Lie groups. For more, see equivalence of definitions of connected algebraic group and equivalence of definitions of connected Lie group.

Facts