= Borel hierarchy
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{title2=$\Sigma^0_\alpha,\Pi^0_\alpha$}
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For a topological space, $\Sigma^0_1$ is the class of open sets and $\Pi^0_\alpha$ consists of complements of $\Sigma^0_\alpha$ sets. For countable $\alpha>1$, $\Sigma^0_\alpha$ consists of countable unions of sets in $\bigcup_{\beta<\alpha}\Pi^0_\beta$. These countable levels contain every <Borel set>. Continuous inverse images preserve each level, by induction using openness, complementation and countable unions.
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