View source for Characteristic property of the product topology/Statement

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==Statement==
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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:
* {{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}}
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==References==
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{{Theorem Of|Topology}}{{Definition|Topology}}
{{Requires references|grade=A|msg=Munkres or Lee's manifolds}}
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=OLD PAGE=
==Statement==
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]])
''if and only if''
* 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.
{{Todo|Diagram}}
==Notes==
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==References==
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{{Theorem Of|Topology}}
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