Proposition 2 (Associativity of topological products)

Let $(X_i)_{i \in I}$ be a family of topological spaces, $(J_x)_{x \in K}$ a partition of the set $I$, and for each $x \in K$ let $X'_x = \prod_{i \in J_x} X_i$ be the product of the spaces $X_i$ for $i \in J_K$. Then the canonical mapping of the product space

is a homeomorphism.

This is a particular case of transitivity of initial topologies ($\S$ 2, no. 3, Proposition 5; cf. Set Theory, Chapter IV, § 2, no. 4, criterion CST 13).

Generally we identify the product spaces $\prod_{i \in I} X_i$ and $\prod_{x \in K} X'_x$ by means of the canonical mapping.