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In mathematics, a Borel set is any set in a topological space that can be formed from open sets and closed sets through the operations of countable union and countable intersection. Borel sets are named after Émile Borel.
For a topological space X, the collection of all Borel sets on X forms a σ-algebra, known as the Borel algebra or Borel σ-algebra. The Borel algebra on X is the smallest σ-algebra containing all open sets (or, equivalently, all closed sets).
Borel sets are important in measure theory, since any measure defined on open sets and closed sets must also be defined on all Borel sets. Any measure defined on the Borel sets is called a Borel measure. Borel sets and the associated Borel hierarchy also play a fundamental role in descriptive set theory.
In some contexts, the Borel sets are defined using compact sets and their complements rather than closed and open sets. These two definitions are equivalent for most typical spaces, including any locally compact, separable metric space (or more generally any σ-compact space), but are different for certain pathological spaces.
For a topological space X, the collection of all Borel sets on X forms a σ-algebra, known as the Borel algebra or Borel σ-algebra. The Borel algebra on X is the smallest σ-algebra containing all open sets (or, equivalently, all closed sets).
Borel sets are important in measure theory, since any measure defined on open sets and closed sets must also be defined on all Borel sets. Any measure defined on the Borel sets is called a Borel measure. Borel sets and the associated Borel hierarchy also play a fundamental role in descriptive set theory.
In some contexts, the Borel sets are defined using compact sets and their complements rather than closed and open sets. These two definitions are equivalent for most typical spaces, including any locally compact, separable metric space (or more generally any σ-compact space), but are different for certain pathological spaces.
In mathematics, a pathological phenomenon is one whose properties are considered atypically bad or counterintuitive.
Often, when the usefulness of a theorem is challenged by counterexamples, defenders of the theorem argue that the exceptions are pathological. A famous case is the Alexander horned sphere, a counterexample showing that topologically embedding the sphere S2 in R3 may fail to "separate the space cleanly", unless an extra condition of tameness is used to suppress possible wild behaviour. See Jordan-Schönflies theorem.
One can therefore say that (particularly in mathematical analysis and set theory) those searching for the "pathological" are like experimentalists, interested in knocking down potential theorems, in contrast to finding general statements widely applicable. Each activity has its role within mathematics.
Often, when the usefulness of a theorem is challenged by counterexamples, defenders of the theorem argue that the exceptions are pathological. A famous case is the Alexander horned sphere, a counterexample showing that topologically embedding the sphere S2 in R3 may fail to "separate the space cleanly", unless an extra condition of tameness is used to suppress possible wild behaviour. See Jordan-Schönflies theorem.
One can therefore say that (particularly in mathematical analysis and set theory) those searching for the "pathological" are like experimentalists, interested in knocking down potential theorems, in contrast to finding general statements widely applicable. Each activity has its role within mathematics.
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