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Generalize upper semicontinuous regularization using Borel Hierachy
Let $X$ be a metric space. Suppose a real-valued function $f:X\rightarrow \mathbb{R}$ is upper semicontinuous class $2$ if for all $c \in \mathbb{R},$ its preimage $f^{-1}(-\infty,c)$ is $F_{\sigma}$.
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