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The set Γ of all open intervals in forms a basis for the Euclidean topology on .. A non-empty family of subsets of a set X that is closed under finite intersections of two or more sets, which is called a π-system on X, is necessarily a base for a topology on X if and only if it covers X.
FGLM algorithm is such a basis conversion algorithm that works only in the zero-dimensional case (where the polynomials have a finite number of complex common zeros) and has a polynomial complexity in the number of common zeros. A basis conversion algorithm that works is the general case is the Gröbner walk algorithm. [4]
The standard topology on R is generated by the open intervals. The set of all open intervals forms a base or basis for the topology, meaning that every open set is a union of some collection of sets from the base. In particular, this means that a set is open if there exists an open interval of non zero radius about every point in the set.
A generator, in category theory, is an object that can be used to distinguish morphisms; In topology, a collection of sets that generate the topology is called a subbase; Generating set of a topological algebra: S is a generating set of a topological algebra A if the smallest closed subalgebra of A containing S is A
The Golomb topology, [2] or relatively prime integer topology, [6] on the set > of positive integers is obtained by taking as a base the collection of all + with , > and and relatively prime. [2] Equivalently, [ 7 ] the subcollection of such sets with the extra condition b < a {\displaystyle b<a} also forms a base for the topology. [ 6 ]
Filters in topology – Use of filters to describe and characterize all basic topological notions and results. Locally convex topological vector space – Vector space with a topology defined by convex open sets; Neighbourhood (mathematics) – Open set containing a given point; Subbase – Collection of subsets that generate a topology
Thus, we can start with a fixed topology and find subbases for that topology, and we can also start with an arbitrary subcollection of the power set ℘ and form the topology generated by that subcollection. We can freely use either equivalent definition above; indeed, in many cases, one of the two conditions is more useful than the other.
In topology and related areas of mathematics, a topological property or topological invariant is a property of a topological space that is invariant under homeomorphisms. Alternatively, a topological property is a proper class of topological spaces which is closed under homeomorphisms.