# Composite of Evaluation Mapping and Projection

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

Let:

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

Then:

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

## Proof

By definition of projection:

 $\ds \forall x \in X, i \in I: \,$ $\ds \map {\paren{pr_i \circ f} } x$ $=$ $\ds \map {pr_i} {\map f x}$ Definition of Composite Mapping $\ds$ $=$ $\ds \map {pr_i} {\family{\map {f_i} x} }$ Definition of Evaluation Mapping $\ds$ $=$ $\ds \map {f_i} x$ Definition of Projection

From Equality of Mappings:

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

$\blacksquare$