Revision as of 00:55, 3 May 2016 (view source) Alec (Talk | contribs) (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...")		Latest revision as of 20:55, 23 September 2016 (view source) Alec (Talk | contribs) (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>
		+	{{Refactor notice|grade=A|msg=The old page was crap}}
		+	__TOC__
		+	==Statement==
		+	</noinclude>{{:Characteristic property of the product topology/Statement/Diagram}}<!--
		+	-->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:
		+	* {{M|f:Y\rightarrow\prod_{\alpha\in I}X_\alpha}} is [[continuous]]
		+	'''{{iff}}'''
		+	* {{M|1=\forall\beta\in I[f_\beta:Y\rightarrow X_\beta\text{ is continuous}]}} - in words, each component function is continuous
		+	{{Todo|Link to diagram}}
		+	<noinclude><div style="clear:both;"></div>
		+	==References==
		+	<references/>
		+	{{Theorem Of|Topology}}{{Definition|Topology}}
		+	{{Requires references|grade=A|msg=Munkres or Lee's manifolds}}
		+	<hr/><br/><hr/><br/><hr/>
		+	=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==

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Latest revision as of 20:55, 23 September 2016

Statement

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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References

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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]:

• A map, [ilmath]f:Y\rightarrow \prod_{\alpha\in I}X_\alpha[/ilmath] is continuous (where [ilmath]\prod_{\alpha\in I}X_\alpha[/ilmath] is imbued with the product topology and [ilmath]\prod[/ilmath] denotes the Cartesian product)

if and only if

• Each component function, [ilmath]f_\alpha:=\pi_\alpha\circ f[/ilmath] is continuous (where [ilmath]\pi_\alpha[/ilmath] denotes one of the canonical projections of the product topology)

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

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TODO: Diagram

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Notes

References

1. ↑ Introduction to Topological Manifolds - John M. Lee