Composite of Evaluation Mapping and Projection

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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:

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

$\blacksquare$