Borel set

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In mathematics, the Borel sets of a topological space are a particular class of "well-behaved" subsets of that space. For example, whereas an arbitrary subset of the real numbers might fail to be Lebesgue measurable, every Borel set of reals is universally measurable. Which sets are Borel can be specified in a number of equivalent ways. Borel sets are named after Émile Borel. The most usual definition goes through the notion of a σ-algebra, which is a collection of subsets of a topological space X {\displaystyle X} that contains both the empty set and the entire set X {\displaystyle X} , and is closed under countable union and complement. Then we can define the Borel σ-algebra over X {\displaystyle X} to be the smallest σ-algebra containing all open sets of X {\displaystyle X} . A Borel subset of...

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