Difference between revisions of "Product topology"

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m (Reverted edits by JessicaBelinda133 (talk) to last revision by Alec)
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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.
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: '''Note: '''{{Note|Very often confused with the [[Box topology]] see [[Product vs box topology]] for details}}
  
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The projection maps are:
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{{Extra Maths}}__TOC__
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==Definition==
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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:
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* 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]]:
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* {{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\} }}
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** 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.
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** {{Yellow Note|For the sake of contrast, the [[Box topology]] has this definition for a basis:<br/><nowiki>
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</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}}
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* 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:
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** {{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)
  
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<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>
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==Characteristic property==
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{{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}}
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This leads to the main property of the product topology, which can best be expressed as a diagram. Will add that later.
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{{Todo|Finish off}}
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{{Todo}}
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{| class="wikitable" border="1"
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|-
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! <math>\begin{xy}
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\xymatrix{
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& \prod_{\beta\in I}X_\beta \ar[d]^{p_i} \\
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Y \ar[ur]^f \ar[r]_{f_i} & X_i
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}\end{xy} </math>
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|-
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| (Commutes {{M|\forall \alpha\in I}})
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|}
 
{{Definition|Topology}}
 
{{Definition|Topology}}

Revision as of 19:51, 11 October 2015

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

Here [ilmath]p_i[/ilmath] denotes the canonical projection, sometimes [ilmath]\pi_i[/ilmath] is used - I avoid using [ilmath]\pi[/ilmath] because it is too similar to [ilmath]\prod[/ilmath] (at least with my handwriting) - I have seen books using both of these conventions

TODO: Finish off

[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])
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