Intersections of Open Sets and Unions of Closed Sets

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The sets and are a subset of the Borel set. The set is the collection of all sets which are a countable intersections of open sets and the set is the collection of all sets which are a countable union of closed sets.[1]


Let X be a topological space whose collection of open sets is denoted and whose collection of closed sets is denoted , then,

The definitions can be extended as follows. Let w be a non-trivial word in the alphabet of length m. Let u be the first m-1 letters in the word and let be the last letter. Then we define,

if and we define

if .

Examples of Sets

  • For any topological space, X, any open set of X is in .
  • Consider under the standard topology. Then the set of irrational numbers is in .
  • Let be a function. Let be the set of points for which is discontinuous. Then the complement of is in .

Properties of Sets

  • A set S is a set if and only if its complement is a set
  • is closed under finite union and countable intersection
  • Lesbague measurable sets can be thought of as the completion of the Borel sets in the following way. A set is Lebesgue measurable iff it differs from a by a set of measure zero. The backward direction is trivial, since sets and sets of measure zero are both measurable. For the forward direction recall the we may find an open set such that . Then if we define , clearly is a set and by monotonicity for all n, so has measure 0.


  1. Gerald B. Folland, Real Analysis: Modern Techniques and Their Applications, Second Edition, §1.2