Definition:Evaluation Mapping (Topology)

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Let $X$ be a topological space.

Let $\family {Y_i}_{i \mathop \in I}$ be an indexed family of topological spaces for some indexing set $I$.

Let $\family {f_i : X \to Y_i}_{i \mathop \in I}$ be an indexed family of continuous mappings.

Let $\ds Y = \prod_{i \mathop \in I} Y_i$ be the product space of $\family {Y_i}_{i \mathop \in I}$.

Let $f : X \to Y$ be the mapping defined by:

$\forall x \in X : \map f x = \family{\map {f_i} x}_{i \mathop \in I}$

Then $f$ is said to be the evaluation mapping induced by the family $\family {f_i}_{i \mathop \in I}$

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