Canonical projections of the product topology

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Definition

Let $\big((X_\alpha,\mathcal{J}_\alpha)\big)_{\alpha\in I}$ 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: $\pi_\beta:(x_\alpha)_{\alpha\in I}\mapsto x_\beta$

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TODO: Add claims, eg continuity and such

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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.

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TODO: Link to category theory

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See also

• Canonical injection of the subspace topology
• Canonical injections of the disjoint union topology