Difference between revisions of "Connected topological group"

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(Definition)
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Definitions (1) and (2) are clearly equivalent.
 
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.
  
 
==Facts==
 
==Facts==
  
 
# [[Connected implies no proper open subgroup]]
 
# [[Connected implies no proper open subgroup]]

Revision as of 21:27, 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.

Facts

  1. Connected implies no proper open subgroup