Products of Products are Homeomorphic to Collapsed Products
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Theorem
Let $I$ be an index set, and for each $i \in I$ let $J_i$ be an index set.
Let the sets $J_i$ be pairwise disjoint.
Let $\ds J = \bigcup_{i \mathop \in I} J_i$
For each $j \in J$, let $X_j$ be a topological space.
Then $\ds \prod_{j \mathop \in J} X_j$ is homeomorphic to $\ds \prod_{i \mathop \in I} \prod_{j \mathop \in J_i} X_j$.
Proof
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