# Characteristic property of the product topology/Statement

From Maths

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## Statement

- [ilmath]f:Y\rightarrow\prod_{\alpha\in I}X_\alpha[/ilmath] is continuous

- [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

## References

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Munkres or Lee's manifolds

# 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.

TODO: Diagram

## Notes

## References