Characteristic property of the product topology/Statement

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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

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TODO: Link to diagram

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References

Grade: A

This page requires references, it is on a to-do list for being expanded with them.

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Munkres or Lee's manifolds

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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:

• 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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