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

Then:
 * $f$ is continuous

Proof
For each $i \in I$, let:
 * $pr_i : Y \to Y_i$ denote the $i$th projection on $Y$

By definition of projection:
 * $\forall x \in X, i \in I : \map {\paren{pr_i \circ f}} x = \map {pr_i} {\map f x} = \map {f_i} x$

From Equality of Mappings:
 * $\forall i \in I : pr_i \circ f = f_i$

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

From Continuous Mapping to Product Space:
 * $f$ is continuous