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In point-set topology, Kuratowski's closure-complement problem asks for the largest number of distinct sets obtainable by repeatedly applying the set operations of closure and complement to a given starting subset of a topological space. The answer is 14. This result was first published by Kazimierz Kuratowski in 1922. [1]
A pure or homogeneous simplicial k-complex is a simplicial complex where every simplex of dimension less than k is a face of some simplex of dimension exactly k. Informally, a pure 1-complex "looks" like it's made of a bunch of lines, a 2-complex "looks" like it's made of a bunch of triangles, etc.
The closure of a subset is the result of a closure operator applied to the subset. The closure of a subset under some operations is the smallest superset that is closed under these operations. It is often called the span (for example linear span ) or the generated set .
Given a separable extension K′ of K, a Galois closure L of K′ is a type of splitting field, and also a Galois extension of K containing K′ that is minimal, in an obvious sense. Such a Galois closure should contain a splitting field for all the polynomials p over K that are minimal polynomials over K of elements of K ′.
Given an abstract simplicial complex X and a vertex in (), its link (,) is a set containing every face such that and {} is a face of X.. In the special case in which X is a 1-dimensional complex (that is: a graph), (,) contains all vertices such that {,} is an edge in the graph; that is, (,) = = the neighborhood of in the graph.
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers.
The algebraic closure of K is also the smallest algebraically closed field containing K, because if M is any algebraically closed field containing K, then the elements of M that are algebraic over K form an algebraic closure of K. The algebraic closure of a field K has the same cardinality as K if K is infinite, and is countably infinite if K ...
The independence complex of G is an ASC whose faces are all independent sets of G (it is the clique complex of the complement graph of G). Clique complexes are the prototypical example of flag complexes. A flag complex is a complex K with the property that every set, all of whose 2-element subsets are faces of K, is itself a face of K. 3.