Topological Evaluation Mapping is Continuous

Theorem

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 evaluation mapping induced by $\family{f_i}_{i \mathop \in I}$.

Then:

$f$ is continuous

Proof

For each $i \in I$, let:

$\pr_i : Y \to Y_i$ denote the $i$th projection on $Y$

From Composite of Evaluation Mapping and Projection:

$\forall i \in I : \pr_i \circ f = f_i$

By assumption:

$\forall i \in I : \pr_i \circ f$ is continuous

From Continuous Mapping to Product Space:

$f$ is continuous

$\blacksquare$