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

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(created stubs, uploaded and linked to image of proof I did quickly on paper. I'll come back to this when I'm happier with the terminology.)
 
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==[[Characteristic property of the product topology/Statement]]==
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==[[Characteristic property of the product topology/Statement|Statement]]==
 
{{:Characteristic property of the product topology/Statement}}
 
{{:Characteristic property of the product topology/Statement}}
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<div style="clear:both;"></div>
 
==Proof==
 
==Proof==
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{{Requires proof}}
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{{Requires proof|grade=A|msg=Due to importance of page, this proof ought to be filled in, I've done it on paper (rough and neat, here's the neat: [[Media:Quick proof of char prop of product top.JPG]])}}
 
==Notes==
 
==Notes==
 
<references group="Note"/>
 
<references group="Note"/>

Latest revision as of 21:06, 23 September 2016

Stub grade: A
This page is a stub
This page is a stub, so it contains little or minimal information and is on a to-do list for being expanded.The message provided is:
Due to importance this page is a high priority
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 topological manifolds. I'll fill it in when I'm more used to the terminology. I'm not happy with it ATM

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


Proof

Grade: A
This page requires one or more proofs to be filled in, it is on a to-do list for being expanded with them.
Please note that this does not mean the content is unreliable. Unless there are any caveats mentioned below the statement comes from a reliable source. As always, Warnings and limitations will be clearly shown and possibly highlighted if very important (see template:Caution et al).
The message provided is:
Due to importance of page, this proof ought to be filled in, I've done it on paper (rough and neat, here's the neat: Media:Quick proof of char prop of product top.JPG)

Notes

References

v • d • e
 
Topology
Overview of the structures studied in topology
Primitives
Global properties of topological spaces
Local properties of topological spaces
Constructs
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



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