Let (S, T ) be a second-countable topological space and (Y, d) any metric space. Show that the Borel σ-algebra in the product S × Y is the product σ-algebra of the Borel σ-algebras in S and in X. Hint: This improves on Proposition 4.1.7. Let V be any open set in S × Y . Let {Um }m≥1 be a countable base for T . For each m and r > 0 let Vmr := {y ∈ Y : for some δ > 0, Um × B(y, r + δ) ⊂ V } where B(y, t ) := {v ∈ Y : d(y, v) <>t }. Show that each Vmr is open in Y and V = lm,n Um × Vm,1/n.