# Baire sets of $X$ possess the required Cartesian product property

Let $X=X_{1}\times X_{2}$ is locally compact space, and define $$E=\{E_{1}\times E_{2}\;|\; E_{i}\; \text{is a Borel set in}\; X_{i}\; ,\; \text{for}\; i=1,2\}$$ Now why the Baire sets of $X$ are in the $\sigma$-algebra generated by $E$? Of course that every Baire set is Borel too so all Baires of $X_{i}$ is a Borel of it too.

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I deleted a straightforward answer based on the Stone-Weierstrass theorem because apparently AmirHosein doesn't assume the Hausdorff condition. Please specify what you mean by local compactness: a compact neighborhood for every point? a base of compact neighborhoods for every point? a base of closed compact neighborhoods for every point? What is your definition of the Baire $\sigma$-algebra? The one generated by the zero sets of continuous functions or the one generated by (closed?) compact $G_\delta$-sets? – Martin Jan 2 '13 at 22:49
– Martin Jan 5 '13 at 15:05