Difference between revisions of "Connected topological group"

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(Alternative definition for a locally connected topological group)
(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.
 
For a [[locally connected topological group]], being connected is equivalent to having no proper open subgroup.
  
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 groups]] and [[equivalence of definitions of connected Lie groups]].
+
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]].
  
 
==Facts==
 
==Facts==
  
 
# [[Connected implies no proper open subgroup]]
 
# [[Connected implies no proper open subgroup]]

Revision as of 21:29, 12 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.

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

  1. Connected implies no proper open subgroup