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Borel sigma-algebra of a product of second-countable spaces

Statement

Let $X, Y$ be second-countable topological spaces and give $X \times Y$ the product topology. Then $B (X \times Y) = B (X) \otimes B (Y) .$

Facts & Assumptions

Given: Second-countable topological spaces $X, Y$ with their product topology and the product sigma-algebra.

[F1]

A second-countable space has an at most countable basis, and every open set is a union of basis elements. (Second countability: an at most countable basis for the topology)

[F2]

Finite products of at most countable sets are at most countable. (A product of two at most countable sets is at most countable)

[F3]

The product topology is generated by coordinate projections; in the binary product its basic open sets are open rectangles. (The product set $\prod_{i \in I} X_{i}$ of functions choosing a point in each factor, the projections, the box topology, and the product topology as the initial topology of the projections; the empty product is a one-point space)

[F4]

The Borel sigma-algebra is generated by the open sets. (The Borel sigma-algebra of a topological space)

[F5]

The product sigma-algebra is generated by measurable rectangles. (The product sigma-algebra and its finite iterates)

Proof

Given: Second-countable spaces $X, Y$ .

1.1givenF1F2F3F4

Choose at most countable bases $U$ for $X$ and $V$ for $Y$ using [F1]. Their product $U \times V$ is at most countable by [F2], and its members $U \times V$ form a basis for $X \times Y$ by [F3]. Each such rectangle lies in $B (X) \otimes B (Y)$ ; every open subset of $X \times Y$ is a union of a subfamily of this countable basis, hence a countable union of those rectangles. By [F4], this proves $B (X \times Y) \subseteq B (X) \otimes B (Y)$ .

2.1givenF3F4F5algebra∎

The coordinate projections $π_{X} : X \times Y \to X$ and $π_{Y} : X \times Y \to Y$ are continuous by the definition of the product topology [F3]. For Borel $A \subseteq X$ and $C \subseteq Y$ , $A \times C = π_{X}^{- 1} (A) \cap π_{Y}^{- 1} (C)$ is Borel in $X \times Y$ . Thus every generator rectangle of the product sigma-algebra [F5] belongs to $B (X \times Y)$ , giving the reverse inclusion.

Borel sigma-algebra of a product of second-countable spaces

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources