Revision as of 07:36, 23 August 2015 (view source) JessicaBelinda133 (Talk | contribs) ← Older edit		Latest revision as of 20:32, 23 September 2016 (view source) Alec (Talk | contribs) (Creating page (scrapped old content). Refactoring started.)
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−	Given a set {{M|X_{\alpha\in I} }} of [[Indexing set|indexed]] [[Topological space|topological spaces]], we define the product topology, denoted <math>\prod_{\alpha\in I}X_\alpha</math> (yes the [[Cartesian product]]) is the coarsest topology such that all the [[Projection map|projection maps]] are continuous.	+	{{Refactor notice|grade=A|As a part of the topology patrol}}
		+	{{Requires references|grade=A|msg=Check Munkres and Topological Manifolds}}
		+	__TOC__
		+	==Definition==
		+	Let {{M|1=\big((X_\alpha,\mathcal{J}_\alpha)\big)_{\alpha\in I} }} be an arbitrary family of [[topological spaces]]. The ''product topology'' is a new topological space defined on the [[set]] {{M|1=\prod_{\alpha\in I}X_\alpha}} (herein we define {{M|1=X:=\prod_{\alpha\in I}X_\alpha}} for notational convenience, where {{M|\prod_{\alpha\in I}X_\alpha}} denotes the [[Cartesian product]] of the family {{M|1=(X_\alpha)_{\alpha\in I} }}) with [[topology]], {{M|\mathcal{J} }} defined as:
		+	* the [[topology generated by a basis|topology generated by the basis {{M|\mathcal{B} }}]], where {{M|\mathcal{B} }} is defined as follows:
		+	** {{M|1=\mathcal{B}:=\left.\left\{\prod_{\alpha\in I}U_\alpha\ \right\vert\ (\forall\beta\in I[U_\beta\in\mathcal{J}_\beta])\wedge\vert\{U_\alpha\ \vert\ \alpha\in I\wedge U_\alpha\neq X_\alpha\}\vert\in\mathbb{N}\right\} }} {{Caution|I need to check this expression}}
		+	*** In words, {{M|\mathcal{B} }} is the set that contains all [[Cartesian product|Cartesian products]] of open sets, {{M|U_\alpha\in\mathcal{J}_\alpha}} given only finitely many of those open sets are ''not'' equal to {{M|X_\alpha}} itself.
		+	We claim:
		+	# {{M|\mathcal{B} }} satisfies the conditions for [[topology generated by a basis|a topology to be generated by a basis]], thus yielding a topology on {{M|X}}, and
		+	# this topology is the unique topology on {{M|X}} for which the [[characteristic property of the product topology|characteristic property]] (see below) holds
		+	==[[Characteristic property of the product topology|Characteristic property]]==
		+	{{:Characteristic property of the product topology/Statement}}
		+	==Notes==
		+	<references group="Note"/>
		+	==References==
		+	<references/>
		+	{{Topology navbox|plain}}
		+	{{Definition|Topology}}{{Theorem Of|Topology}}
−	The projection maps are:	+	<hr/><br/><hr/><br/><hr/>
		+	=2nd generation page=
		+	: '''Note: ''' for finite collections of topological spaces the product and [[box topology]] agree. In general however the box topology ''does not'' satisfy the [[characteristic property of the product topology]].
		+	__TOC__
		+	==Definition==
		+	Given an arbitrary family of [[topological space|topological spaces]], {{M|\big((X_\alpha,\mathcal{J}_\alpha)\big)_{\alpha\in I} }} the ''product topology'' is a [[topology]] defined on the set {{M|\prod_{\alpha\in I}X_\alpha}} (where {{M|\prod}} denotes the [[Cartesian product]]) to be the [[topology generated by a basis|topology generated by the basis]]:
		+	* {{MM|1=\mathcal{B}:=\left\{\left.\prod_{\alpha\in I}U_\alpha\right\vert\ (U_\alpha)_{\alpha\in I}\in\prod_{\alpha\in I}\mathcal{J}_\alpha\ \wedge\ \Big\vert\{U_\alpha\vert\ U_\alpha\ne X_\alpha\}\Big\vert\in\mathbb{N}\right\} }}
		+	The family of functions, {{M|1=\left\{\pi_\alpha:\prod_{\beta\in I}X_\beta\rightarrow X_\alpha\text{ given by }\pi_\alpha:(x_\gamma)_{\gamma\in I}\mapsto x_\alpha\ \Big\vert\ \alpha\in I\right\} }} are called the ''canonical projections'' for the product.
		+	: '''Claim 1: ''' this is a [[basis for a topology]],
		+	: '''Claim 2: ''' the canonical projections are [[continuous]]
		+	==[[Characteristic property of the product topology|Characteristic property]]==
		+	{{:Characteristic property of the product topology/Statement}}
−	<math>p_\alpha:\prod_{\beta\in I}X_\beta\rightarrow X_\alpha</math> which take the [[Tuple|tuple]] <math>(x_\alpha)_{\alpha\in I}\rightarrow x_{\beta}</math>	+	=OLD PAGE=
		+	: '''Note: '''{{Note|Very often confused with the [[Box topology]] see [[Product vs box topology]] for details}}
−	This leads to the main property of the product topology, which can best be expressed as a diagram. As shown below:	+	{{Extra Maths}}__TOC__
		+	==Definition==
		+	Given an arbitrary collection of [[Indexing set|indexed]] {{M|(X_\alpha,\mathcal{J}_\alpha)_{\alpha\in I} }} [[Topological space|topological spaces]], we define the '''product topology''' as follows:
		+	* Let {{M|1=X:=\prod_{\alpha\in I}X_\alpha}} be a set imbued with the [[Topological space|topology]] generated by the [[Basis (topology)|basis]]:
		+	* {{M|1=\mathcal{B}=\left\{\prod_{\alpha\in I}U_\alpha\Big\vert\ \forall\alpha\in I[U_\alpha\in\mathcal{J}_\alpha]\wedge\exists n\in\mathbb{N}[\vert\{U_\alpha\vert U_\alpha\ne X_\alpha\}\vert=n]\right\} }}
		+	** That is to say the basis set contains all the products of open sets where the product has a finite number of elements that are not the entirety of their space.
		+	** {{Yellow Note|For the sake of contrast, the [[Box topology]] has this definition for a basis:<br/><nowiki>
		+	</nowiki>{{M|1=\mathcal{B}_\text{box}=\left\{\prod_{\alpha\in I}U_\alpha\Big\vert\ \forall\alpha\in I[U_\alpha\in\mathcal{J}_\alpha]\right\} }} - the product of any collection of open sets}}
		+	* Note that in the case of a finite number of spaces, say {{M|1=(X_i,\mathcal{J}_i)_{i=1}^n}} then the topology on {{M|1=\prod_{i=1}^nX_i}} is generated by the basis:
		+	** {{M|1=\mathcal{B}_\text{finite}=\left\{\prod^n_{i=1}U_i\Big\vert\ \forall i\in\{1,2,\ldots,n\}[U_i\in\mathcal{J}_i]\right\} }} (that is to say the box/product topologies agree)
−	10 x 5 = 50! this is a product
−		+	==Characteristic property==
−	{{Todo}}	+	{{Yellow Note|Here {{M|p_i}} denotes the ''[[canonical projection]]'', sometimes {{M|\pi_i}} is used - I avoid using {{M|\pi}} because  it is too similar to {{M|\prod}} (at least with my handwriting) - I have seen books using both of these conventions}}
		+	{{Todo|Finish off}}
		+	{| class="wikitable" border="1"
		+	|-
		+	! <math>\begin{xy}
		+	\xymatrix{
		+	& \prod_{\beta\in I}X_\beta \ar[d]^{p_i} \\
		+	Y \ar[ur]^f \ar[r]_{f_i} & X_i
		+	}\end{xy} </math>
		+	|-
		+	| (Commutes {{M|\forall \alpha\in I}})
		+	|}
	{{Definition|Topology}}		{{Definition|Topology}}

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Latest revision as of 20:32, 23 September 2016

Definition

Let [ilmath]\big((X_\alpha,\mathcal{J}_\alpha)\big)_{\alpha\in I}[/ilmath] be an arbitrary family of topological spaces. The product topology is a new topological space defined on the set [ilmath]\prod_{\alpha\in I}X_\alpha[/ilmath] (herein we define [ilmath]X:=\prod_{\alpha\in I}X_\alpha[/ilmath] for notational convenience, where [ilmath]\prod_{\alpha\in I}X_\alpha[/ilmath] denotes the Cartesian product of the family [ilmath](X_\alpha)_{\alpha\in I}[/ilmath]) with topology, [ilmath]\mathcal{J} [/ilmath] defined as:

• the topology generated by the basis [ilmath]\mathcal{B} [/ilmath], where [ilmath]\mathcal{B} [/ilmath] is defined as follows:
  • [ilmath]\mathcal{B}:=\left.\left\{\prod_{\alpha\in I}U_\alpha\ \right\vert\ (\forall\beta\in I[U_\beta\in\mathcal{J}_\beta])\wedge\vert\{U_\alpha\ \vert\ \alpha\in I\wedge U_\alpha\neq X_\alpha\}\vert\in\mathbb{N}\right\}[/ilmath] Caution:I need to check this expression
    • In words, [ilmath]\mathcal{B} [/ilmath] is the set that contains all Cartesian products of open sets, [ilmath]U_\alpha\in\mathcal{J}_\alpha[/ilmath] given only finitely many of those open sets are not equal to [ilmath]X_\alpha[/ilmath] itself.

We claim:

1. [ilmath]\mathcal{B} [/ilmath] satisfies the conditions for a topology to be generated by a basis, thus yielding a topology on [ilmath]X[/ilmath], and
2. this topology is the unique topology on [ilmath]X[/ilmath] for which the characteristic property (see below) holds

Characteristic property

Let [ilmath]\big((X_\alpha,\mathcal{J}_\alpha)\big)_{\alpha\in I} [/ilmath] be an arbitrary family of topological spaces and let [ilmath](Y,\mathcal{ K })[/ilmath] be a topological space. Consider [ilmath](\prod_{\alpha\in I}X_\alpha,\mathcal{J})[/ilmath] as a topological space with topology ([ilmath]\mathcal{J} [/ilmath]) given by the product topology of [ilmath]\big((X_\alpha,\mathcal{J}_\alpha)\big)_{\alpha\in I} [/ilmath]. Lastly, let [ilmath]f:Y\rightarrow\prod_{\alpha\in I}X_\alpha[/ilmath] be a map, and for [ilmath]\alpha\in I[/ilmath] define [ilmath]f_\alpha:Y\rightarrow X_\alpha[/ilmath] as [ilmath]f_\alpha=\pi_\alpha\circ f[/ilmath] (where [ilmath]\pi_\alpha[/ilmath] denotes the [ilmath]\alpha^\text{th} [/ilmath] canonical projection of the product topology) then:

• [ilmath]f:Y\rightarrow\prod_{\alpha\in I}X_\alpha[/ilmath] is continuous

if and only if

• [ilmath]\forall\beta\in I[f_\beta:Y\rightarrow X_\beta\text{ is continuous}][/ilmath] - in words, each component function is continuous

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TODO: Link to diagram

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Notes

References

v • d • e   Topology Overview of the structures studied in topology Primitives topological space, open set, neighbourhood Global properties of topological spaces Compactness, Connectedness (Path-Connectedness), Hausdorff-ness Local properties of topological spaces Constructs product (see also box, difference between), subspace (AKA: induced), quotient, disjoint union, adjunction, cone over a topological space Subtypes of topological spaces inner product spaces [ilmath]\subset[/ilmath] normed spaces [ilmath]\subset[/ilmath] metric spaces [ilmath]\subset[/ilmath] topological spaces Maps between topological spaces Continuous map (AKA: topological homomorphism), topological homeomorphism

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2nd generation page

Note: for finite collections of topological spaces the product and box topology agree. In general however the box topology does not satisfy the characteristic property of the product topology.

Definition

Given an arbitrary family of topological spaces, [ilmath]\big((X_\alpha,\mathcal{J}_\alpha)\big)_{\alpha\in I} [/ilmath] the product topology is a topology defined on the set [ilmath]\prod_{\alpha\in I}X_\alpha[/ilmath] (where [ilmath]\prod[/ilmath] denotes the Cartesian product) to be the topology generated by the basis:

• [math]\mathcal{B}:=\left\{\left.\prod_{\alpha\in I}U_\alpha\right\vert\ (U_\alpha)_{\alpha\in I}\in\prod_{\alpha\in I}\mathcal{J}_\alpha\ \wedge\ \Big\vert\{U_\alpha\vert\ U_\alpha\ne X_\alpha\}\Big\vert\in\mathbb{N}\right\}[/math]

The family of functions, [ilmath]\left\{\pi_\alpha:\prod_{\beta\in I}X_\beta\rightarrow X_\alpha\text{ given by }\pi_\alpha:(x_\gamma)_{\gamma\in I}\mapsto x_\alpha\ \Big\vert\ \alpha\in I\right\}[/ilmath] are called the canonical projections for the product.

Claim 1: this is a basis for a topology,

Claim 2: the canonical projections are continuous

Characteristic property

Let [ilmath]\big((X_\alpha,\mathcal{J}_\alpha)\big)_{\alpha\in I} [/ilmath] be an arbitrary family of topological spaces and let [ilmath](Y,\mathcal{ K })[/ilmath] be a topological space. Consider [ilmath](\prod_{\alpha\in I}X_\alpha,\mathcal{J})[/ilmath] as a topological space with topology ([ilmath]\mathcal{J} [/ilmath]) given by the product topology of [ilmath]\big((X_\alpha,\mathcal{J}_\alpha)\big)_{\alpha\in I} [/ilmath]. Lastly, let [ilmath]f:Y\rightarrow\prod_{\alpha\in I}X_\alpha[/ilmath] be a map, and for [ilmath]\alpha\in I[/ilmath] define [ilmath]f_\alpha:Y\rightarrow X_\alpha[/ilmath] as [ilmath]f_\alpha=\pi_\alpha\circ f[/ilmath] (where [ilmath]\pi_\alpha[/ilmath] denotes the [ilmath]\alpha^\text{th} [/ilmath] canonical projection of the product topology) then:

• [ilmath]f:Y\rightarrow\prod_{\alpha\in I}X_\alpha[/ilmath] is continuous

if and only if

• [ilmath]\forall\beta\in I[f_\beta:Y\rightarrow X_\beta\text{ is continuous}][/ilmath] - in words, each component function is continuous

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TODO: Link to diagram

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OLD PAGE

Note: Very often confused with the Box topology see Product vs box topology for details

[math]\newcommand{\bigudot}{ \mathchoice{\mathop{\bigcup\mkern-15mu\cdot\mkern8mu}}{\mathop{\bigcup\mkern-13mu\cdot\mkern5mu}}{\mathop{\bigcup\mkern-13mu\cdot\mkern5mu}}{\mathop{\bigcup\mkern-13mu\cdot\mkern5mu}} }[/math][math]\newcommand{\udot}{\cup\mkern-12.5mu\cdot\mkern6.25mu\!}[/math][math]\require{AMScd}\newcommand{\d}[1][]{\mathrm{d}^{#1} }[/math]

Definition

Given an arbitrary collection of indexed [ilmath](X_\alpha,\mathcal{J}_\alpha)_{\alpha\in I} [/ilmath] topological spaces, we define the product topology as follows:

• Let [ilmath]X:=\prod_{\alpha\in I}X_\alpha[/ilmath] be a set imbued with the topology generated by the basis:
• [ilmath]\mathcal{B}=\left\{\prod_{\alpha\in I}U_\alpha\Big\vert\ \forall\alpha\in I[U_\alpha\in\mathcal{J}_\alpha]\wedge\exists n\in\mathbb{N}[\vert\{U_\alpha\vert U_\alpha\ne X_\alpha\}\vert=n]\right\}[/ilmath]
  • That is to say the basis set contains all the products of open sets where the product has a finite number of elements that are not the entirety of their space.
  • For the sake of contrast, the Box topology has this definition for a basis:
    [ilmath]\mathcal{B}_\text{box}=\left\{\prod_{\alpha\in I}U_\alpha\Big\vert\ \forall\alpha\in I[U_\alpha\in\mathcal{J}_\alpha]\right\}[/ilmath] - the product of any collection of open sets
• Note that in the case of a finite number of spaces, say [ilmath](X_i,\mathcal{J}_i)_{i=1}^n[/ilmath] then the topology on [ilmath]\prod_{i=1}^nX_i[/ilmath] is generated by the basis:
  • [ilmath]\mathcal{B}_\text{finite}=\left\{\prod^n_{i=1}U_i\Big\vert\ \forall i\in\{1,2,\ldots,n\}[U_i\in\mathcal{J}_i]\right\}[/ilmath] (that is to say the box/product topologies agree)

Characteristic property

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TODO: Finish off

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[math]\begin{xy} \xymatrix{ & \prod_{\beta\in I}X_\beta \ar[d]^{p_i} \\ Y \ar[ur]^f \ar[r]_{f_i} & X_i }\end{xy} [/math]
(Commutes [ilmath]\forall \alpha\in I[/ilmath])