n. 1. the smallest closed set containing a given set, equal to the intersection of all closed sets containing that set. For example, the closure of the positive integers under subtraction is the set of all integers. See closed (sense 1). See also hull, algebraic closure. 2. the set of points in a space every neighborhood of which has a non-empty intersection with a given set. The closure of A is written A or Cl A. For example, the closure of the open interval (0, 1) is the closed interval [0, 1]; the closure of the rationals is the reals. Compare interior. See also cluster point. 3. (Logic) the closed sentence formed by prefixing quantifiers to a given open sentence to bind all its free variables; especially the universal closure of the given sentence, formed by binding its free variables with universal quantifiers. Mathematical identities written without quantifiers are abbreviations for their universal closures; thus one writes a + b = b + a as the commutative law for addition, indicating that the result of adding any two elements is independent of order. 4. the operation forming such a set or sentence.