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Let $\{ X_t : t \in T\}$, be a family of topological spaces. Suppose thst $T = \bigcup \{ T_s : s \in S \}$, where $T_s \neq \emptyset $ for all $s \in S$, and $T_s \cap T_{s'} = \emptyset$ if $s \neq s'$.

How can I prove that $\Pi_{t \in T} X_t$ and $\Pi_{s \in S}(\Pi_{t \in T_s} X_t)$ are homeomorphic.

Intuitively, it seems rather obvious. But, the formal proof, unfortunately, I am not sure where to start..

Any help?

Thank you!

topsi
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    Take the natural bijection between the spaces, and show it is continuous in both directions using the universal/characteristic property of topological products. – Daniel Fischer Apr 09 '14 at 08:12

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