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This article lists mathematical properties and laws of sets, involving the set-theoretic operations of union, intersection, and complementation and the relations of set equality and set inclusion. It also provides systematic procedures for evaluating expressions, and performing calculations, involving these operations and relations.
For example, the union of three sets A, B, and C contains all elements of A, all elements of B, and all elements of C, and nothing else. Thus, x is an element of A ∪ B ∪ C if and only if x is in at least one of A, B, and C. A finite union is the union of a finite number of sets; the phrase does not imply that the union set is a finite set ...
The open sets and closed sets of any topological space are closed under both unions and intersections. [ 1 ] On the real line R , the family of sets consisting of the empty set and all finite unions of half-open intervals of the form ( a , b ] , with a , b ∈ R is a ring in the measure-theoretic sense.
It is the algebra of the set-theoretic operations of union, intersection and complementation, and the relations of equality and inclusion. For a basic introduction to sets see the article on sets, for a fuller account see naive set theory, and for a full rigorous axiomatic treatment see axiomatic set theory.
The Cantor–Bendixson theorem states that any Polish space can be written as the union of a countable set and a perfect set. Because any G δ subset of a Polish space is again a Polish space, the theorem also shows that any G δ subset of a Polish space is the union of a countable set and a set that is perfect with respect to the induced topology.
Any union of elements of τ is an element of τ; Any intersection of finitely many elements of τ is an element of τ; If τ is a topology on X, then the pair (X, τ) is called a topological space. The notation X τ may be used to denote a set X endowed with the particular topology τ. The members of τ are called open sets in X.
A topological space consists of a pair (,) where is a set (whose elements are called points) and is a topology on , which is a family of sets (whose elements are called open sets) over that contains both the empty set and itself, and is closed under arbitrary set unions and finite set intersections.
The answer seems to be every possible . When is empty, the condition given above is an example of a vacuous truth. So the intersection of the empty family should be the universal set (the identity element for the operation of intersection), [4] but in standard set theory, the universal set does not exist.