Suppose $X$ is a compact metric space and $A\subset X$ a subset such that $A\times A$ a Borel subset of $X\times X$. Is $A$ necessarily Borel?
2026-03-29 11:02:09.1774782129
If $A\times A$ is Borel must $A$ be Borel?
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Define $f:X\to X\times X$ by $f(x)=(x,x)$. Then $f$ is continuous, and $f^{-1}(A\times A)=A$. If $A\times A$ is Borel, it follows that $A$ is Borel.