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In mathematics, a set A is a subset of a set B if all elements of A are also elements of B; B is then a superset of A. It is possible for A and B to be equal; if they are unequal, then A is a proper subset of B. The relationship of one set being a subset of another is called inclusion (or sometimes containment).
The first definition is usually taken for locally compact, countably compact, metrizable, separable, countable; the second for locally connected. [15] Locally closed subset A subset of a topological space that is the intersection of an open and a closed subset. Equivalently, it is a relatively open subset of its closure. Locally compact
Consequently, the term greatest lower bound (abbreviated as GLB) is also commonly used. [1] The supremum (abbreviated sup; pl.: suprema) of a subset of a partially ordered set is the least element in that is greater than or equal to each element of , if such an element exists. [1]
In topology and related areas of mathematics, a subset A of a topological space X is said to be dense in X if every point of X either belongs to A or else is arbitrarily "close" to a member of A — for instance, the rational numbers are a dense subset of the real numbers because every real number either is a rational number or has a rational number arbitrarily close to it (see Diophantine ...
Subspace, a particular subset of a parent space; A subset of a topological space endowed with the subspace topology; Linear subspace, in linear algebra, a subset of a vector space that is closed under addition and scalar multiplication
In set theory and related branches of mathematics, a family (or collection) can mean, depending upon the context, any of the following: set, indexed set, multiset, or class. A collection of subsets of a given set is called a family of subsets of , or a family of sets over .
The only subset of the empty set is the empty set itself; equivalently, the power set of the empty set is the set containing only the empty set. The number of elements of the empty set (i.e., its cardinality) is zero. The empty set is the only set with either of these properties. For any set A: The empty set is a subset of A
In mathematics, a subset of a given set is closed under an operation of the larger set if performing that operation on members of the subset always produces a member of that subset. For example, the natural numbers are closed under addition, but not under subtraction: 1 − 2 is not a natural number, although both 1 and 2 are.