Characteristic property of the product topology/Statement

Statement

Let $\big((X_\alpha,\mathcal{J}_\alpha)\big)_{\alpha\in I}$ be an arbitrary family of topological spaces. Let $(Y,\mathcal{ K })$ be any topological space. Then:

• A map, $f:Y\rightarrow \prod_{\alpha\in I}X_\alpha$ is continuous (where $\prod_{\alpha\in I}X_\alpha$ is imbued with the product topology and $\prod$ denotes the Cartesian product)

if and only if

• Each component function, $f_\alpha:=\pi_\alpha\circ f$ is continuous (where $\pi_\alpha$ denotes one of the canonical projections of the product topology)

Furthermore, the product topology is the unique topology on $\prod_{\alpha\in I}X_\alpha$ with this property.

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TODO: Diagram

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