Search results
Results From The WOW.Com Content Network
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.
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.
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 ]
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
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
Magma contains asymptotically fast algorithms for all fundamental dense matrix operations, such as Strassen multiplication. Sparse matrices Magma contains the structured Gaussian elimination and Lanczos algorithms for reducing sparse systems which arise in index calculus methods, while Magma uses Markowitz pivoting for several other sparse ...
In the following, represents the real numbers with their usual topology. The subspace topology of the natural numbers, as a subspace of , is the discrete topology.; The rational numbers considered as a subspace of do not have the discrete topology ({0} for example is not an open set in because there is no open subset of whose intersection with can result in only the singleton {0}).
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.