Semilocally simply connected space - Revision history

Vipul: /* Definition */

2012-01-28T01:00:58Z

Definition

← Older revision		Revision as of 01:00, 28 January 2012
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	# For any <math>x \in X</math> there exists an open subset <math>U \ni x</math> such that the homomorphism of fundamental groups induced by the inclusion of <math>U</math> in <math>X</math> is trivial (i.e., the image of the inclusion is the trivial subgroup), the inclusion being: <math>\! \pi_1(U,x) \to \pi_1(X,x)</math>. Note that if <math>U</math> and/or <math>X</math> are not connected, we interpret the fundamental groups as referring to the fundamental groups of the [[path component]]s of <math>x</math> in the respective subsets.		# For any <math>x \in X</math> there exists an open subset <math>U \ni x</math> such that the homomorphism of fundamental groups induced by the inclusion of <math>U</math> in <math>X</math> is trivial (i.e., the image of the inclusion is the trivial subgroup), the inclusion being: <math>\! \pi_1(U,x) \to \pi_1(X,x)</math>. Note that if <math>U</math> and/or <math>X</math> are not connected, we interpret the fundamental groups as referring to the fundamental groups of the [[path component]]s of <math>x</math> in the respective subsets.
	# For any <math>x \in X</math> and any open subset <math>V</math> of <math>X</math> containing <math>x</math>, there exists an open subset <math>U \ni x</math> such that <math>U \subseteq V</math> and the homomorphism of fundamental groups induced by the inclusion of <math>U</math> in <math>X</math> is trivial (i.e., the image of the inclusion is the trivial subgroup), the inclusion being: <math>\! \pi_1(U,x) \to \pi_1(X,x)</math>. In other words, every loop about <math>x</math> contained in <math>U</math>, is nullhomotopic in <math>X</math>. Note that if <math>U</math> and/or <math>X</math> are not connected, we interpret the fundamental groups as referring to the fundamental groups of the [[path component]]s of <math>x</math> in <math>U</math> and <math>X</math> respectively.		# For any <math>x \in X</math> and any open subset <math>V</math> of <math>X</math> containing <math>x</math>, there exists an open subset <math>U \ni x</math> such that <math>U \subseteq V</math> and the homomorphism of fundamental groups induced by the inclusion of <math>U</math> in <math>X</math> is trivial (i.e., the image of the inclusion is the trivial subgroup), the inclusion being: <math>\! \pi_1(U,x) \to \pi_1(X,x)</math>. In other words, every loop about <math>x</math> contained in <math>U</math>, is nullhomotopic in <math>X</math>. Note that if <math>U</math> and/or <math>X</math> are not connected, we interpret the fundamental groups as referring to the fundamental groups of the [[path component]]s of <math>x</math> in <math>U</math> and <math>X</math> respectively.

			Note that the term is typically used for spaces that are [[locally path-connected space]]s. In this case, we can assume that the open subset <math>U</math> is path-connected.

	==Relation with other properties==		==Relation with other properties==

Vipul: /* Definition */

2012-01-28T00:59:49Z

Definition

← Older revision		Revision as of 00:59, 28 January 2012
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	A [[topological space]] is said to be '''semilocally simply connected''' or '''semilocally 1-connected''' if it satisfies the following equivalent conditions:		A [[topological space]] is said to be '''semilocally simply connected''' or '''semilocally 1-connected''' if it satisfies the following equivalent conditions:

	# For any <math>x \in X</math> there exists an open subset <math>U \ni x</math> such that the homomorphism of fundamental groups induced by the inclusion of <math>U</math> in <math>X</math> is trivial (i.e., the image of the inclusion is the trivial subgroup), the inclusion being: <math>\! \pi_1(U,x) \to \pi_1(X,x)</math>. Note that if <math>U<math> and/or <math>X</math> are not connected, we interpret the fundamental groups as referring to the fundamental groups of the [[path component]]s of <math>x</math> in the respective subsets.		# For any <math>x \in X</math> there exists an open subset <math>U \ni x</math> such that the homomorphism of fundamental groups induced by the inclusion of <math>U</math> in <math>X</math> is trivial (i.e., the image of the inclusion is the trivial subgroup), the inclusion being: <math>\! \pi_1(U,x) \to \pi_1(X,x)</math>. Note that if <math>U</math> and/or <math>X</math> are not connected, we interpret the fundamental groups as referring to the fundamental groups of the [[path component]]s of <math>x</math> in the respective subsets.
	# For any <math>x \in X</math> and any open subset <math>V</math> of <math>X</math> containing <math>x</math>, there exists an open subset <math>U \ni x</math> such that <math>U \subseteq V</math> and the homomorphism of fundamental groups induced by the inclusion of <math>U</math> in <math>X</math> is trivial (i.e., the image of the inclusion is the trivial subgroup), the inclusion being: <math>\! \pi_1(U,x) \to \pi_1(X,x)</math>. In other words, every loop about <math>x</math> contained in <math>U</math>, is nullhomotopic in <math>X</math>. Note that if <math>U</math> and/or <math>X</math> are not connected, we interpret the fundamental groups as referring to the fundamental groups of the [[path component]]s of <math>x</math> in <math>U</math> and <math>X</math> respectively.		# For any <math>x \in X</math> and any open subset <math>V</math> of <math>X</math> containing <math>x</math>, there exists an open subset <math>U \ni x</math> such that <math>U \subseteq V</math> and the homomorphism of fundamental groups induced by the inclusion of <math>U</math> in <math>X</math> is trivial (i.e., the image of the inclusion is the trivial subgroup), the inclusion being: <math>\! \pi_1(U,x) \to \pi_1(X,x)</math>. In other words, every loop about <math>x</math> contained in <math>U</math>, is nullhomotopic in <math>X</math>. Note that if <math>U</math> and/or <math>X</math> are not connected, we interpret the fundamental groups as referring to the fundamental groups of the [[path component]]s of <math>x</math> in <math>U</math> and <math>X</math> respectively.

Vipul: /* Definition */

2012-01-28T00:59:35Z

Definition

← Older revision		Revision as of 00:59, 28 January 2012
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	# For any <math>x \in X</math> there exists an open subset <math>U \ni x</math> such that the homomorphism of fundamental groups induced by the inclusion of <math>U</math> in <math>X</math> is trivial (i.e., the image of the inclusion is the trivial subgroup), the inclusion being: <math>\! \pi_1(U,x) \to \pi_1(X,x)</math>. Note that if <math>U<math> and/or <math>X</math> are not connected, we interpret the fundamental groups as referring to the fundamental groups of the [[path component]]s of <math>x</math> in the respective subsets.		# For any <math>x \in X</math> there exists an open subset <math>U \ni x</math> such that the homomorphism of fundamental groups induced by the inclusion of <math>U</math> in <math>X</math> is trivial (i.e., the image of the inclusion is the trivial subgroup), the inclusion being: <math>\! \pi_1(U,x) \to \pi_1(X,x)</math>. Note that if <math>U<math> and/or <math>X</math> are not connected, we interpret the fundamental groups as referring to the fundamental groups of the [[path component]]s of <math>x</math> in the respective subsets.
	# For any <math>x \in X</math> and any open subset <math>V</math> of <math>X</math> containing <math>x</math>, there exists an open subset <math>U \ni x</math> such that <math>U \subseteq V</math> and the homomorphism of fundamental groups induced by the inclusion of <math>U</math> in <math>X</math> is trivial (i.e., the image of the inclusion is the trivial subgroup), the inclusion being: <math>\! \pi_1(U,x) \to \pi_1(X,x)</math>. In other words, every loop about <math>x</math> contained in <math>U</math>, is nullhomotopic in <math>X</math>. Note that if <math>U<math> and/or <math>X</math> are not connected, we interpret the fundamental groups as referring to the fundamental groups of the [[path component]]s of <math>x</math> in <math>U</math> and <math>X</math> respectively.		# For any <math>x \in X</math> and any open subset <math>V</math> of <math>X</math> containing <math>x</math>, there exists an open subset <math>U \ni x</math> such that <math>U \subseteq V</math> and the homomorphism of fundamental groups induced by the inclusion of <math>U</math> in <math>X</math> is trivial (i.e., the image of the inclusion is the trivial subgroup), the inclusion being: <math>\! \pi_1(U,x) \to \pi_1(X,x)</math>. In other words, every loop about <math>x</math> contained in <math>U</math>, is nullhomotopic in <math>X</math>. Note that if <math>U</math> and/or <math>X</math> are not connected, we interpret the fundamental groups as referring to the fundamental groups of the [[path component]]s of <math>x</math> in <math>U</math> and <math>X</math> respectively.

	==Relation with other properties==		==Relation with other properties==

Vipul at 00:58, 28 January 2012

2012-01-28T00:58:38Z

← Older revision		Revision as of 00:58, 28 January 2012
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	==Definition==		==Definition==

	~~===Symbol~~-~~free definition===~~		A [[topological space]] is said to be '''semilocally simply connected''' or '''semilocally 1-connected''' if it satisfies the following equivalent conditions:

	~~A [[topological space]] is said to be '''semilocally simply connected''' or '''semilocally 1-connected''' if every point~~ in ~~the space has~~ an open ~~neighbourhood~~ such that the inclusion ~~map from~~ that ~~neighbourhood~~ to the ~~whoel space induces a trivial mapping at the level of~~ fundamental groups.		# For any <math>x \in X</math> there exists an open subset <math>U \ni x</math> such that the homomorphism of fundamental groups induced by the inclusion of <math>U</math> in <math>X</math> is trivial (i.e., the image of the inclusion is the trivial subgroup), the inclusion being: <math>\! \pi_1(U,x) \to \pi_1(X,x)</math>. Note that if <math>U<math> and/or <math>X</math> are not connected, we interpret the fundamental groups as referring to the fundamental groups of the [[path component]]s of <math>x</math> in the respective subsets.
			# For any <math>x \in X</math> and any open subset <math>V</math> of <math>X</math> containing <math>x</math>, there exists an open subset <math>U \ni x</math> such that <math>U \subseteq V</math> and the homomorphism of fundamental groups induced by the inclusion of <math>U</math> in <math>X</math> is trivial (i.e., the image of the inclusion is the trivial subgroup), the inclusion being: <math>\! \pi_1(U,x) \to \pi_1(X,x)</math>. In other words, every loop about <math>x</math> contained in <math>U</math>, is nullhomotopic in <math>X</math>. Note that if <math>U<math> and/or <math>X</math> are not connected, we interpret the fundamental groups as referring to the fundamental groups of the [[path component]]s of <math>x</math> in <math>U</math> and <math>X</math> respectively.
	~~===Definition with symbols===~~

	A [[~~topological space~~]] <math>X</math> ~~is said to be '''semilocally simply connected''' if for~~ any <math>x \in X</math> ~~there exists a neighbourhood~~ <math>U</math> of <math>x</math> such that the homomorphism of fundamental groups induced by the inclusion of <math>U</math> in <math>X</math>, is trivial, the inclusion being:

	<math>\pi_1(U,x) \to \pi_1(X,x)</math>

	In other words, every loop about <math>x</math> contained in <math>U</math>, is nullhomotopic in <math>X</math>.

	==Relation with other properties==		==Relation with other properties==

Vipul: 3 revisions

2008-05-11T19:58:00Z

3 revisions

← Older revision	Revision as of 19:58, 11 May 2008
(No difference)

Vipul at 21:54, 21 April 2008

2008-04-21T21:54:38Z

← Older revision		Revision as of 21:54, 21 April 2008
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	===Symbol-free definition===		===Symbol-free definition===

	A [[topological space]] is said to be '''semilocally simply connected''' if every point in the space has an open neighbourhood such that the inclusion map from that neighbourhood to the whoel space induces a trivial mapping at the level of fundamental groups.		A [[topological space]] is said to be '''semilocally simply connected''' or '''semilocally 1-connected''' if every point in the space has an open neighbourhood such that the inclusion map from that neighbourhood to the whoel space induces a trivial mapping at the level of fundamental groups.

	===Definition with symbols===		===Definition with symbols===

	A [[topological space]] <math>X</math> is said to be '''semilocally simply connected''' if for any <math>x \in X</math> there exists a neighbourhood <math>U</math> of <math>x</math> such that the homomorphism of fundamental groups induced by the inclusion of <math>U</math> in <math>X</math>, is trivial. In other words, every loop about <math>x</math> contained in <math>U</math>, is nullhomotopic in <math>X</math>.		A [[topological space]] <math>X</math> is said to be '''semilocally simply connected''' if for any <math>x \in X</math> there exists a neighbourhood <math>U</math> of <math>x</math> such that the homomorphism of fundamental groups induced by the inclusion of <math>U</math> in <math>X</math>, is trivial, the inclusion being:

			<math>\pi_1(U,x) \to \pi_1(X,x)</math>

			In other words, every loop about <math>x</math> contained in <math>U</math>, is nullhomotopic in <math>X</math>.

	==Relation with other properties==		==Relation with other properties==
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	* [[Locally simply connected space]]		* [[Locally simply connected space]]
	* [[Simply connected space]]		* [[Simply connected space]]

			==References==
			===Textbook references===
			* {{booklink\|Munkres}}, Page 494 (formal definition)
			* {{booklink\|Rotman}}, Page 297 (formal definition): Introduced as '''semilocally 1-connected'''
			* {{booklink\|Hatcher}}, Page 63 (formal definition)
			* {{booklink\|Spanier}}, Page 78 (forma definition): Introduced as '''semilocally 1-connected'''

Vipul at 21:35, 30 September 2007

2007-09-30T21:35:40Z

← Older revision		Revision as of 21:35, 30 September 2007
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	A [[topological space]] is said to be '''semilocally simply connected''' if every point in the space has an open neighbourhood such that the inclusion map from that neighbourhood to the whoel space induces a trivial mapping at the level of fundamental groups.		A [[topological space]] is said to be '''semilocally simply connected''' if every point in the space has an open neighbourhood such that the inclusion map from that neighbourhood to the whoel space induces a trivial mapping at the level of fundamental groups.

			===Definition with symbols===

			A [[topological space]] <math>X</math> is said to be '''semilocally simply connected''' if for any <math>x \in X</math> there exists a neighbourhood <math>U</math> of <math>x</math> such that the homomorphism of fundamental groups induced by the inclusion of <math>U</math> in <math>X</math>, is trivial. In other words, every loop about <math>x</math> contained in <math>U</math>, is nullhomotopic in <math>X</math>.

	==Relation with other properties==		==Relation with other properties==

Vipul at 03:14, 22 May 2007

2007-05-22T03:14:38Z

New page

{{homotopy-invariant topospace property}}

==Definition==

===Symbol-free definition===

A [[topological space]] is said to be '''semilocally simply connected''' if every point in the space has an open neighbourhood such that the inclusion map from that neighbourhood to the whoel space induces a trivial mapping at the level of fundamental groups.

==Relation with other properties==

===Stronger properties===

* [[Locally simply connected space]]
* [[Simply connected space]]