Composite of Evaluation Mapping and Projection

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


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


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


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:

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