Examples
- In topology and related branches, the relevant operation is taking limits. The topological closure of a set is the corresponding closure operator. The Kuratowski closure axioms characterize this operator.
- In linear algebra, the linear span of a set X of vectors is the closure of that set; it is the smallest subset of the vector space that includes X and is closed under the operation of linear combination. This subset is a subspace.
- In matroid theory, the closure of X is the largest superset of X that has the same rank as X.
- In set theory, the transitive closure of a set.
- In set theory, the transitive closure of a binary relation.
- In algebra, the algebraic closure of a field.
- In commutative algebra, closure operations for ideals, as integral closure and tight closure.
- In geometry, the convex hull of a set S of points is the smallest convex set of which S is a subset.
- In the theory of formal languages, the Kleene closure of a language can be described as the set of strings that can be made by concatenating zero or more strings from that language.
- In group theory, the conjugate closure or normal closure of a set of group elements is the smallest normal subgroup containing the set.
Read more about this topic: Closure (mathematics)
Famous quotes containing the word examples:
“In the examples that I here bring in of what I have [read], heard, done or said, I have refrained from daring to alter even the smallest and most indifferent circumstances. My conscience falsifies not an iota; for my knowledge I cannot answer.”
—Michel de Montaigne (1533–1592)
“It is hardly to be believed how spiritual reflections when mixed with a little physics can hold people’s attention and give them a livelier idea of God than do the often ill-applied examples of his wrath.”
—G.C. (Georg Christoph)
“Histories are more full of examples of the fidelity of dogs than of friends.”
—Alexander Pope (1688–1744)