Difference between revisions of "Connected topological group"

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# It is connected as a topological space.
 
# 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.
 
  
 
===Equivalence of definitions===
 
===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.
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Definitions (1) and (2) are clearly equivalent.
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==Facts==
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# [[Connected implies no proper open subgroup]]

Revision as of 21:13, 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.

Facts

  1. Connected implies no proper open subgroup