Partition of Indexing Set induces Bijection on Family of Sets

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Theorem

Let $I$ be an indexing set.

Let $\family {S_\alpha}_{\alpha \mathop \in I}$ be a family of sets indexed by $I$.

Let $\family {I_\gamma}_{\gamma \mathop \in J}$ be a partitioning of $I$.


Then there exists a bijection:

$\ds \phi: \prod_{\gamma \mathop \in J} \paren {\prod_{\alpha \mathop \in I_\gamma} S_\alpha} \to \prod_{\alpha \mathop \in I} S_\alpha$


Proof

First a lemma:

Let $I$ be an indexing set.

Let $\family {S_\alpha}_{\alpha \mathop \in I}$ be a family of sets indexed by $I$.

Let $I = I_1 \cup I_2$ such that $I_1 \cap I_2 = \O$.


Then there exists a bijection:

$\ds \psi: \paren {\prod_{\alpha \mathop \in I_1} S_\alpha} \times \paren {\prod_{\alpha \mathop \in I_2} S_\alpha} \to \prod_{\alpha \mathop \in I} S_\alpha$

$\Box$


We can define a projection $\pr_\gamma$:

$\ds \pr_\gamma: \prod_{\gamma \mathop \in J} \paren {\prod_{\alpha \mathop \in I_\gamma} S_\alpha} \to \prod_{\alpha \mathop \in I_\gamma} S_\alpha$

so that for $\ds X \in \prod_{\gamma \mathop \in J} \paren {\prod_{\alpha \mathop \in I_\gamma} S_\alpha}, X = \family {X_\gamma}_{\gamma \mathop \in J}$:

$\map {\pr_\gamma} X = X_\gamma$


From the lemma we can build $\phi$ so that for $I_\gamma = \set {\alpha: \alpha \in I_\gamma}$:

$\ds \map \phi {\mathbf x} = \prod_{\gamma \mathop \in J} \paren {\prod_{\alpha \mathop \in I_\gamma} \map {\psi_\alpha} {x_\alpha} }$

as $\phi$ uniquely sets all components of $\map {\pr_\gamma} X$ for all $\gamma$.

The result follows.

$\blacksquare$


Sources

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