Subspace of Product Space is Homeomorphic to Factor Space/Proof 1/Lemma 2
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Theorem
Let $\family {X_i}_{i \mathop \in I}$ be a family of sets where $I$ is an arbitrary index set.
Let $\ds X = \prod_{i \mathop \in I} X_i$ be the Cartesian product of $\family {X_i}_{i \mathop \in I}$.
Let $z \in X$.
Let $i \in I$.
Let $\pr_i : X \to X_i$ be the $i$th-projection from $X$.
For all for all $j \in I$ let:
- $Z_j = \begin{cases} X_i & i = j \\ \set{z_j} & j \ne i \end{cases}$
Let $Y_i = \prod_{j \mathop \in I} Z_j$
Let $p_i : Y_i \to X_i$ be the $i$th-projection from $Y_i$.
Then:
- $\pr_i {\restriction_{Y_i} } = p_i$
Proof
For all $y \in Y_i$:
\(\ds \map {\pr_i {\restriction_{Y_i} } } y\) | \(=\) | \(\ds \map {\pr_i} y\) | Definition of Restriction of Mapping: $\pr_i {\restriction_{Y_i} } : Y_i \to X_i$ | |||||||||||
\(\ds \) | \(=\) | \(\ds y_i\) | Definition of Projection: $\pr_i: X \to X_i$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \map {p_i} y\) | Definition of Projection: $p_i: Y_i \to X_i$ |
- $\pr_i {\restriction_{Y_i} } = p_i$
$\blacksquare$