Canonical projections of the product topology

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Let [ilmath]\big((X_\alpha,\mathcal{J}_\alpha)\big)_{\alpha\in I} [/ilmath] be an arbitrary family of topological spaces and let [ilmath](\prod_{\alpha\in I}X_\alpha,\mathcal{J})[/ilmath] denote their product, considered with the product topology, then, for each [ilmath]\beta\in I[/ilmath] we get a map:

  • [ilmath]\pi_\beta:\prod_{\alpha\in I}X_\alpha\rightarrow X_\beta[/ilmath] given by: [ilmath]\pi_\beta:(x_\alpha)_{\alpha\in I}\mapsto x_\beta[/ilmath]

TODO: Add claims, eg continuity and such

Sometimes denoted by [ilmath]p_\beta:\prod_{\alpha\in I}X_\alpha\rightarrow X_\beta[/ilmath] instead. We'll use the two interchangeably but will always define them as a canonical projection.

TODO: Link to category theory

See also