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.
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.
The topology generated by ... are a basis for the usual Euclidean topology. ... are the product topology, where the family of functions is the set of ...
The Vietoris topology on the set of all non-empty subsets of a topological space , named for Leopold Vietoris, is generated by the following basis: for every -tuple , …, of open sets in , we construct a basis set consisting of all subsets of the union of the that have non-empty intersections with each .
The finest topology on X is the discrete topology; this topology makes all subsets open. The coarsest topology on X is the trivial topology; this topology only admits the empty set and the whole space as open sets. In function spaces and spaces of measures there are often a number of possible topologies.
Though the subspace topology of Y = {−1} ∪ {1/n } n∈N in the section above is shown not to be generated by the induced order on Y, it is nonetheless an order topology on Y; indeed, in the subspace topology every point is isolated (i.e., singleton {y} is open in Y for every y in Y), so the subspace topology is the discrete topology on Y (the topology in which every subset of Y is open ...
When the basis {b n} is normalized, the coordinate functionals {b* n} have norm ≤ 2C in the continuous dual V ′ of V. Since every vector v in a Banach space V with a Schauder basis is the limit of P n (v), with P n of finite rank and uniformly bounded, such a space V satisfies the bounded approximation property.
If the codomain of the functions under consideration has a uniform structure or a metric structure then the compact-open topology is the "topology of uniform convergence on compact sets." That is to say, a sequence of functions converges in the compact-open topology precisely when it converges uniformly on every compact subset of the domain. [2]