The Borel hierarchy is a classification of certain sets in a topological space, particularly in the context of the real numbers and standard Borel spaces. This hierarchy ranks sets based on their complexity in terms of open and closed sets. The Borel hierarchy is crucial in descriptive set theory, a branch of mathematical logic and set theory dealing with the study of definable subsets of Polish spaces (completely metrizable separable topological spaces).
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For a topological space, is the class of open sets and consists of complements of sets. For countable , consists of countable unions of sets in . These countable levels contain every Borel set. Continuous inverse images preserve each level, by induction using openness, complementation and countable unions.