Products of Products are Homeomorphic to Collapsed Products

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 $\ds \prod_{i \mathop \in I} \prod_{j \mathop \in J_i} X_j$.