Definition:Evaluation Mapping (Topology)

Definition

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 $\family {f_i}_{i \mathop \in I}$

Also see

• Every Topological Evaluation Mapping is Continuous

Sources

• 1955: John L. Kelley: General Topology: Chapter $4$: Embedding and Metrization, $\S$Embedding in Cubes
• 1970: Stephen Willard: General Topology: Chapter $3$: New Space from Old: $\S8$: Product Spaces, Weak Topologies: Definition $8.11$