Canonical projections of the product topology
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[hide]Definition
Let ((Xα,Jα))α∈I be an arbitrary family of topological spaces and let (\prod_{\alpha\in I}X_\alpha,\mathcal{J}) denote their product, considered with the product topology, then, for each \beta\in I we get a map:
- \pi_\beta:\prod_{\alpha\in I}X_\alpha\rightarrow X_\beta given by: \pi_\beta:(x_\alpha)_{\alpha\in I}\mapsto x_\beta
TODO: Add claims, eg continuity and such
Sometimes denoted by p_\beta:\prod_{\alpha\in I}X_\alpha\rightarrow X_\beta instead. We'll use the two interchangeably but will always define them as a canonical projection.
TODO: Link to category theory
See also
References
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