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

Revision as of 00:29, 3 May 2016 (view source) Alec (Talk | contribs) (Creating skeleton for page.)		Latest revision as of 21:06, 23 September 2016 (view source) Alec (Talk | contribs) (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]]==	+	{{Stub page|grade=A|msg=Due to importance this page is a high priority}}
		+	{{Requires references|grade=A|msg=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}}
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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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	==Proof==		==Proof==
−	{{Requires proof}}	+	{{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"/>

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Latest revision as of 21:06, 23 September 2016

Statement

Let $\big((X_\alpha,\mathcal{J}_\alpha)\big)_{\alpha\in I}$ be an arbitrary family of topological spaces and let $(Y,\mathcal{ K })$ be a topological space. Consider $(\prod_{\alpha\in I}X_\alpha,\mathcal{J})$ as a topological space with topology ($\mathcal{J}$) given by the product topology of $\big((X_\alpha,\mathcal{J}_\alpha)\big)_{\alpha\in I}$. Lastly, let $f:Y\rightarrow\prod_{\alpha\in I}X_\alpha$ be a map, and for $\alpha\in I$ define $f_\alpha:Y\rightarrow X_\alpha$ as $f_\alpha=\pi_\alpha\circ f$ (where $\pi_\alpha$ denotes the $\alpha^\text{th}$ canonical projection of the product topology) then:

• $f:Y\rightarrow\prod_{\alpha\in I}X_\alpha$ is continuous

if and only if

• $\forall\beta\in I[f_\beta:Y\rightarrow X_\beta\text{ is continuous}]$ - in words, each component function is continuous

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TODO: Link to diagram

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Proof

Notes

References

v • d • e   Topology Overview of the structures studied in topology Primitives topological space, open set, neighbourhood Global properties of topological spaces Compactness, Connectedness (Path-Connectedness), Hausdorff-ness Local properties of topological spaces Constructs product (see also box, difference between), subspace (AKA: induced), quotient, disjoint union, adjunction, cone over a topological space Subtypes of topological spaces inner product spaces $\subset$ normed spaces $\subset$ metric spaces $\subset$ topological spaces Maps between topological spaces Continuous map (AKA: topological homomorphism), topological homeomorphism