Connected topological group - Revision history

Vipul: /* Alternative definition for a locally connected topological group */

2012-01-13T18:03:02Z

Alternative definition for a locally connected topological group

← Older revision		Revision as of 18:03, 13 January 2012
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	===Alternative definition for a locally connected topological group===		===Alternative definition for a locally connected topological group===

	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. See [[locally connected and no proper open subgroup implies connected]]

	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]].		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]].

Vipul: /* Facts */

2012-01-12T21:30:22Z

Facts

← Older revision		Revision as of 21:30, 12 January 2012
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	==Facts==		==Facts==

	# [[Connected implies no proper open subgroup]]		* [[Connected implies no proper open subgroup]]: The converse holds for [[locally connected topological group]]s.

Vipul: /* Alternative definition for a locally connected topological group */

2012-01-12T21:29:58Z

Alternative definition for a locally connected topological group

← Older revision		Revision as of 21:29, 12 January 2012
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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]]

Vipul: /* Alternative definition for a locally connected topological group */

2012-01-12T21:29:41Z

Alternative definition for a locally connected topological group

← Older revision		Revision as of 21:29, 12 January 2012
Line 17:		Line 17:

	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]].

	==Facts==		==Facts==

	# [[Connected implies no proper open subgroup]]		# [[Connected implies no proper open subgroup]]

Vipul: /* Definition */

2012-01-12T21:27:35Z

Definition

← Older revision		Revision as of 21:27, 12 January 2012
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	Definitions (1) and (2) are clearly equivalent.		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==		==Facts==

	# [[Connected implies no proper open subgroup]]		# [[Connected implies no proper open subgroup]]

Vipul at 21:13, 12 January 2012

2012-01-12T21:13:25Z

← Older revision		Revision as of 21:13, 12 January 2012
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	# It is connected as a topological space.		# It is connected as a topological space.
	# ~~It has no proper~~ [[~~open subgroup]]~~		# 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.~~		Definitions (1) and (2) are clearly equivalent.

			==Facts==

			# [[Connected implies no proper open subgroup]]

Vipul at 19:41, 22 June 2008

2008-06-22T19:41:22Z

← Older revision		Revision as of 19:41, 22 June 2008
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	==Definition==		==Definition==

			===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.		# It is connected as a topological space.
	* It has no proper [[open subgroup]]		# It has no proper [[open subgroup]]
	* The connected component of the identity element equals the whole group.		# 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.

Vipul: 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...

2008-06-22T19:37:54Z

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...

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 space.
* It has no proper [[open subgroup]]
* The connected component of the identity element equals the whole group.