Difference between revisions of "Characteristic property of the product topology/Statement"

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(Created page with "<noinclude> ==Statement== </noinclude> Let {{M|\big((X_\alpha,\mathcal{J}_\alpha)\big)_{\alpha\in I} }} be an arbitrary family of topological spaces. Let {{Top.|Y|K}} be a...")
 
(Refactored the page to make it less crap. Included diagram, totally restated the property, although I'm not 100% happy with it)
 
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<noinclude>
 
<noinclude>
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{{Refactor notice|grade=A|msg=The old page was crap}}
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__TOC__
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==Statement==
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</noinclude>{{:Characteristic property of the product topology/Statement/Diagram}}<!--
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-->Let {{M|\big((X_\alpha,\mathcal{J}_\alpha)\big)_{\alpha\in I} }} be an arbitrary family of [[topological spaces]] and let {{Top.|Y|K}} be a topological space. Consider {{M|(\prod_{\alpha\in I}X_\alpha,\mathcal{J})}} as a topological space with [[topology]] ({{M|\mathcal{J} }}) given by the [[product topology|product topology of {{M|\big((X_\alpha,\mathcal{J}_\alpha)\big)_{\alpha\in I} }}]]. Lastly, let {{M|f:Y\rightarrow\prod_{\alpha\in I}X_\alpha}} be a [[map]], and for {{M|\alpha\in I}} define {{M|1=f_\alpha:Y\rightarrow X_\alpha}} as {{M|1=f_\alpha=\pi_\alpha\circ f}} (where {{M|\pi_\alpha}} denotes the {{M|\alpha^\text{th} }} [[canonical projection of the product topology]]) then:
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* {{M|f:Y\rightarrow\prod_{\alpha\in I}X_\alpha}} is [[continuous]]
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'''{{iff}}'''
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* {{M|1=\forall\beta\in I[f_\beta:Y\rightarrow X_\beta\text{ is continuous}]}} - in words, each component function is continuous
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{{Todo|Link to diagram}}
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<noinclude><div style="clear:both;"></div>
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==References==
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<references/>
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{{Theorem Of|Topology}}{{Definition|Topology}}
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{{Requires references|grade=A|msg=Munkres or Lee's manifolds}}
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<hr/><br/><hr/><br/><hr/>
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=OLD PAGE=
 
==Statement==
 
==Statement==
</noinclude>
 
 
Let {{M|\big((X_\alpha,\mathcal{J}_\alpha)\big)_{\alpha\in I} }} be an arbitrary family of [[topological spaces]]. Let {{Top.|Y|K}} be any [[topological space]]. Then{{rITTMJML}}:
 
Let {{M|\big((X_\alpha,\mathcal{J}_\alpha)\big)_{\alpha\in I} }} be an arbitrary family of [[topological spaces]]. Let {{Top.|Y|K}} be any [[topological space]]. Then{{rITTMJML}}:
 
* A [[map]], {{M|f:Y\rightarrow \prod_{\alpha\in I}X_\alpha}} is [[continuous]] (where {{M|\prod_{\alpha\in I}X_\alpha}} is imbued with the [[product topology]] and {{M|\prod}} denotes the [[Cartesian product]])
 
* A [[map]], {{M|f:Y\rightarrow \prod_{\alpha\in I}X_\alpha}} is [[continuous]] (where {{M|\prod_{\alpha\in I}X_\alpha}} is imbued with the [[product topology]] and {{M|\prod}} denotes the [[Cartesian product]])
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* Each component function, {{M|1=f_\alpha:=\pi_\alpha\circ f}} is [[continuous]] (where {{M|\pi_\alpha}} denotes one of the [[canonical projections of the product topology]])
 
* Each component function, {{M|1=f_\alpha:=\pi_\alpha\circ f}} is [[continuous]] (where {{M|\pi_\alpha}} denotes one of the [[canonical projections of the product topology]])
 
Furthermore, the [[product topology]] is the unique topology on {{M|\prod_{\alpha\in I}X_\alpha}} with this property.
 
Furthermore, the [[product topology]] is the unique topology on {{M|\prod_{\alpha\in I}X_\alpha}} with this property.
<noinclude>
 
 
{{Todo|Diagram}}
 
{{Todo|Diagram}}
 
==Notes==
 
==Notes==

Latest revision as of 20:55, 23 September 2016

Grade: A
This page is currently being refactored (along with many others)
Please note that this does not mean the content is unreliable. It just means the page doesn't conform to the style of the site (usually due to age) or a better way of presenting the information has been discovered.
The message provided is:
The old page was crap

Statement

[ilmath]\begin{xy} \xymatrix{ & & \prod_{\alpha\in I}X_\alpha \ar[dd] \\ & & \\ Y \ar[uurr]^f \ar[rr]+<-0.9ex,0.15ex>|(.875){\hole} & & X_b \save (15,13)+"3,3"*+{\ldots}="udots"; (8.125,6.5)+"3,3"*+{X_a}="x1"; (-8.125,-6.5)+"3,3"*+{X_c}="x3"; (-15,-13)+"3,3"*+{\ldots}="ldots"; \ar@{->} "x1"; "1,3"; \ar@{->}_(0.65){\pi_c,\ \pi_b,\ \pi_a} "x3"; "1,3"; \ar@{->}|(.873){\hole} "x1"+<-0.9ex,0.15ex>; "3,1"; \ar@{->}_{f_c,\ f_b,\ f_a} "x3"+<-0.9ex,0.3ex>; "3,1"; \restore } \end{xy}[/ilmath]

TODO: Caption


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

TODO: Link to diagram


References

Grade: A
This page requires references, it is on a to-do list for being expanded with them.
Please note that this does not mean the content is unreliable, it just means that the author of the page doesn't have a book to hand, or remember the book to find it, which would have been a suitable reference.
The message provided is:
Munkres or Lee's manifolds





OLD PAGE

Statement

Let [ilmath]\big((X_\alpha,\mathcal{J}_\alpha)\big)_{\alpha\in I} [/ilmath] be an arbitrary family of topological spaces. Let [ilmath](Y,\mathcal{ K })[/ilmath] be any topological space. Then[1]:

if and only if

Furthermore, the product topology is the unique topology on [ilmath]\prod_{\alpha\in I}X_\alpha[/ilmath] with this property.


TODO: Diagram


Notes

References

  1. Introduction to Topological Manifolds - John M. Lee