Subspace topology - Revision history

Vipul: /* Definition */

2012-01-24T23:28:23Z

Definition

← Older revision		Revision as of 23:28, 24 January 2012
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	Let <math>(X,\tau)</math> be a topological space (viz, a set <math>X</math> endowed with a topology <math>\tau</math>) and <math>A</math> be a subset of <math>X</math>. The ''subspace topology'' or ''induced topology'' or ''relative topology'' on <math>A</math> can be defined in many equivalent ways. Note that <math>A</math> induced with this topology is a topological space in its own right. Thus, subsets of topological spaces are often also called subspaces.		Let <math>(X,\tau)</math> be a topological space (viz, a set <math>X</math> endowed with a topology <math>\tau</math>) and <math>A</math> be a subset of <math>X</math>. The ''subspace topology'' or ''induced topology'' or ''relative topology'' on <math>A</math> can be defined in many equivalent ways. Note that <math>A</math> induced with this topology is a topological space in its own right. Thus, subsets of topological spaces are often also called subspaces.

	~~===Definition~~ in ~~open set terms===~~		The subspace topology can be defined in many equivalent ways. The equivalent formulations are described below:

	~~The subspace topology on <math>~~A~~</math> is as follows: a~~ subset of <math>A</math> is open in <math>A</math> if and only if it is the intersection with <math>A</math> of an [[open subset]] of <math>X</math>. In other words, <math>U</math> is open in <math>A</math> if and only if there exists an open subset <math>V</math> of <math>X</math> such that <math>U = V \cap A</math>.		{\| class="sortable" border="1"
			! Formulation in terms of ... !! Details !! Comment
	~~(Note that~~ <math>V</math> is not uniquely determined by <math>U</math>).		\|-
			\| determining which subsets are open \|\| A subset of <math>A</math> is open in <math>A</math> if and only if it is the intersection with <math>A</math> of an [[open subset]] of <math>X</math>. In other words, <math>U</math> is open in <math>A</math> if and only if there exists an open subset <math>V</math> of <math>X</math> such that <math>U = V \cap A</math>. \|\| The description of which subsets are open completely determines the topology, and the topology completely determines which subsets are open.<br>Also, <math>V</math> is not uniquely determined by <math>U</math>, though, subject to its existence, we can take a ''maximal'' <math>V</math>, which is the union of all the possible choices for <math>V</math>.
	~~===Definition in closed set terms===~~		\|-
			\| determining which subsets are closed \|\| A subset of <math>A</math> is closed in <math>A</math> if and only if it is the intersection with <math>A</math> of a [[closed subset]] of <math>X</math>. In other words, <math>C</math> is closed in <math>A</math> if and only if there exists a closed subset <math>D</math> of <math>X</math> such that <math>C = D \cap A</math>. \|\| The description of which subsets are closed completely determines the topology, and the topology completely determines which subsets are open.<br>Also, <math>D</math> is not uniquely determined by <math>C</math>, though, subject to its existence, we can take a ''minimal'' <math>D</math>, which is the intersection of all possible choices, and is also the closure of <math>C</math> within <math>X</math>.
	~~The subspace topology on~~ <math>A</math> is ~~as follows: a~~ subset of <math>A</math> is closed in <math>A</math> if and only if it is the intersection with <math>A</math> of a [[closed subset]] of <math>X</math>. In other words, <math>C</math> is closed in <math>A</math> if and only if there exists a closed subset <math>D</math> of <math>X</math> such that <math>C = D \cap A</math>.		\|-
			\| finding a basis \|\| Given a [[basis]] for <math>\tau</math>, we can directly use it to define a basis for the subspace topology on <math>A</math>. Namely, for each basis open set, replace it by its intersection with <math>A</math>. In other words, if <math>\{ U_i \}_{i \in I}</math> form a basis for <math>\tau</math>, then a basis for the subspace topology on <math>A</math> is given by <math>\{ U_i \cap A \}_{i \in I}</math>. \|\| Note that while a basis uniquely determines the topology, the same topology can be describd by different possibilities for basis. Thus, ''a priori'', it is not clear that different starting choices of basis for <math>X</math> would yield the same topology on <math>A</math>. To show that this is true, we show equivalence with the open subset formulation.
	~~(Note that~~ <math>D</math> is not uniquely determined by <math>C</math>).		\|-
			\| finding a subbasis \|\| Given a [[subbasis]] for <math>\tau</math>, we can directly use it to define a basis for the subspace topology on <math>A</math>. Namely, for each subbasis open set, replace it by its intersection with <math>A</math>. In other words, if <math>\{ U_i \}_{i \in I}</math> form a subbasis for <math>\tau</math>, then a basis for the subspace topology on <math>A</math> is given by <math>\{ U_i \cap A \}_{i \in I}</math>. \|\| Note that while a subbasis uniquely determines the topology, the same topology can be described by different possibilities for basis. Thus, ''a priori'', it is not clear that different starting choices of basis for <math>X</math> would yield the same topology on <math>A</math>. To show that this is true, we show equivalence with the open subset formulation.
	~~===Definition in~~ basis ~~terms===~~		\|-
			\| in terms of making a map continuous \|\| The subspace topology is the [[coarsest topology]] that can be endowed on <math>A</math>, for which the inclusion map from <math>A</math> to <math>X</math> is a [[continuous map]]. \|\|
	Given a [[basis]] for <math>\tau</math>, we can directly use it to define a basis for the subspace topology on <math>A</math>. Namely, for each basis open set, replace it by its intersection with <math>A</math>.		\|}

	In other words, if <math>\{ U_i \}_{i \in I}</math> form a basis for <math>\tau</math>, then a basis for the subspace topology on <math>A</math> is given by <math>\{ U_i \cap A \}_{i \in I}</math>.

	~~===Definition in~~ subbasis ~~terms===~~

	Given a [[subbasis]] for <math>\tau</math>, we can directly use it to define a basis for the subspace topology on <math>A</math>. Namely, for each subbasis open set, replace it by its intersection with <math>A</math>.

	In other words, if <math>\{ U_i \}_{i \in I}</math> form a subbasis for <math>\tau</math>, then a basis for the subspace topology on <math>A</math> is given by <math>\{ U_i \cap A \}_{i \in I}</math>.

	~~===Definition~~ in terms of ~~maps===~~

	The subspace topology is the [[coarsest topology]] that can be endowed to <math>A</math>, for which the inclusion map from <math>A</math> to <math>X</math>, is a [[continuous map]].

	===Equivalence of definitions===		===Equivalence of definitions===

	{{further\|[[Equivalence of definitions of subspace topology]]}}		{{further\|[[Equivalence of definitions of subspace topology]]}}

	==References==		==References==
	===Textbook references===		===Textbook references===
	* {{booklink\|Munkres}}, Page 88-89		* {{booklink\|Munkres}}, Page 88-89
	* {{booklink\|SingerThorpe}}, Page 10 (Theorem 6): introduced under the name '''relative topology'''		* {{booklink\|SingerThorpe}}, Page 10 (Theorem 6): introduced under the name '''relative topology'''

Vipul at 01:22, 25 November 2008

2008-11-25T01:22:39Z

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	{{basicdef}}		{{basicdef}}
			{{induced subspace structure\|topological space}}
	==Definition==		==Definition==

Vipul at 21:01, 19 July 2008

2008-07-19T21:01:06Z

← Older revision		Revision as of 21:01, 19 July 2008
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	===Definition in open set terms===		===Definition in open set terms===

	The subspace topology on <math>A</math> is as follows: a subset of <math>A</math> is open in <math>A</math> if and only if it is the intersection with <math>A</math> of an [[open subset]] of <math>X</math>.		The subspace topology on <math>A</math> is as follows: a subset of <math>A</math> is open in <math>A</math> if and only if it is the intersection with <math>A</math> of an [[open subset]] of <math>X</math>. In other words, <math>U</math> is open in <math>A</math> if and only if there exists an open subset <math>V</math> of <math>X</math> such that <math>U = V \cap A</math>.

			(Note that <math>V</math> is not uniquely determined by <math>U</math>).

	===Definition in closed set terms===		===Definition in closed set terms===

	The subspace topology on <math>A</math> is as follows: a subset of <math>A</math> is closed in <math>A</math> if and only if it is the intersection with <math>A</math> of a [[closed subset]] of <math>X</math>.		The subspace topology on <math>A</math> is as follows: a subset of <math>A</math> is closed in <math>A</math> if and only if it is the intersection with <math>A</math> of a [[closed subset]] of <math>X</math>. In other words, <math>C</math> is closed in <math>A</math> if and only if there exists a closed subset <math>D</math> of <math>X</math> such that <math>C = D \cap A</math>.

			(Note that <math>D</math> is not uniquely determined by <math>C</math>).

	===Definition in basis terms===		===Definition in basis terms===

	Given a [[basis]] for <math>\tau</math>, we can directly use it to define a basis for the subspace topology on <math>A</math>. Namely, for each basis open set, replace it by its intersection with <math>A</math>.		Given a [[basis]] for <math>\tau</math>, we can directly use it to define a basis for the subspace topology on <math>A</math>. Namely, for each basis open set, replace it by its intersection with <math>A</math>.

			In other words, if <math>\{ U_i \}_{i \in I}</math> form a basis for <math>\tau</math>, then a basis for the subspace topology on <math>A</math> is given by <math>\{ U_i \cap A \}_{i \in I}</math>.

	===Definition in subbasis terms===		===Definition in subbasis terms===

	Given a [[subbasis]] for <math>\tau</math>, we can directly use it to define a basis for the subspace topology on <math>A</math>. Namely, for each subbasis open set, replace it by its intersection with <math>A</math>.		Given a [[subbasis]] for <math>\tau</math>, we can directly use it to define a basis for the subspace topology on <math>A</math>. Namely, for each subbasis open set, replace it by its intersection with <math>A</math>.

			In other words, if <math>\{ U_i \}_{i \in I}</math> form a subbasis for <math>\tau</math>, then a basis for the subspace topology on <math>A</math> is given by <math>\{ U_i \cap A \}_{i \in I}</math>.

	===Definition in terms of maps===		===Definition in terms of maps===
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	The subspace topology is the [[coarsest topology]] that can be endowed to <math>A</math>, for which the inclusion map from <math>A</math> to <math>X</math>, is a [[continuous map]].		The subspace topology is the [[coarsest topology]] that can be endowed to <math>A</math>, for which the inclusion map from <math>A</math> to <math>X</math>, is a [[continuous map]].

			===Equivalence of definitions===

			{{further\|[[Equivalence of definitions of subspace topology]]}}
	==References==		==References==
	===Textbook references===		===Textbook references===
	* {{booklink\|Munkres}}, Page 88-89		* {{booklink\|Munkres}}, Page 88-89
	* {{booklink\|SingerThorpe}}, Page 10 (Theorem 6): introduced under the name '''relative topology'''		* {{booklink\|SingerThorpe}}, Page 10 (Theorem 6): introduced under the name '''relative topology'''

Vipul: 3 revisions

2008-05-11T19:59:22Z

3 revisions

← Older revision	Revision as of 19:59, 11 May 2008
(No difference)

Vipul at 20:54, 21 April 2008

2008-04-21T20:54:40Z

← Older revision		Revision as of 20:54, 21 April 2008
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	The subspace topology is the [[coarsest topology]] that can be endowed to <math>A</math>, for which the inclusion map from <math>A</math> to <math>X</math>, is a [[continuous map]].		The subspace topology is the [[coarsest topology]] that can be endowed to <math>A</math>, for which the inclusion map from <math>A</math> to <math>X</math>, is a [[continuous map]].

			==References==
			===Textbook references===
			* {{booklink\|Munkres}}, Page 88-89
			* {{booklink\|SingerThorpe}}, Page 10 (Theorem 6): introduced under the name '''relative topology'''

Vipul at 23:05, 26 December 2007

2007-12-26T23:05:07Z

← Older revision		Revision as of 23:05, 26 December 2007
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			{{basicdef}}

	==Definition==		==Definition==

Vipul at 09:32, 20 August 2007

2007-08-20T09:32:05Z

New page

==Definition==

Let <math>(X,\tau)</math> be a topological space (viz, a set <math>X</math> endowed with a topology <math>\tau</math>) and <math>A</math> be a subset of <math>X</math>. The ''subspace topology'' or ''induced topology'' or ''relative topology'' on <math>A</math> can be defined in many equivalent ways. Note that <math>A</math> induced with this topology is a topological space in its own right. Thus, subsets of topological spaces are often also called subspaces.

===Definition in open set terms===

The subspace topology on <math>A</math> is as follows: a subset of <math>A</math> is open in <math>A</math> if and only if it is the intersection with <math>A</math> of an [[open subset]] of <math>X</math>.

===Definition in closed set terms===

The subspace topology on <math>A</math> is as follows: a subset of <math>A</math> is closed in <math>A</math> if and only if it is the intersection with <math>A</math> of a [[closed subset]] of <math>X</math>.

===Definition in basis terms===

Given a [[basis]] for <math>\tau</math>, we can directly use it to define a basis for the subspace topology on <math>A</math>. Namely, for each basis open set, replace it by its intersection with <math>A</math>.

===Definition in subbasis terms===

Given a [[subbasis]] for <math>\tau</math>, we can directly use it to define a basis for the subspace topology on <math>A</math>. Namely, for each subbasis open set, replace it by its intersection with <math>A</math>.

===Definition in terms of maps===

The subspace topology is the [[coarsest topology]] that can be endowed to <math>A</math>, for which the inclusion map from <math>A</math> to <math>X</math>, is a [[continuous map]].