Search results
Results From The WOW.Com Content Network
In mathematics, a base (or basis; pl.: bases) for the topology ... The topology is called the topology generated by , and is called a subbase for . The ...
The second subbase generates the usual topology as well, since the open intervals (,) with , rational, are a basis for the usual Euclidean topology. The subbase consisting of all semi-infinite open intervals of the form ( − ∞ , a ) {\displaystyle (-\infty ,a)} alone, where a {\displaystyle a} is a real number, does not generate the usual ...
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 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 Golomb topology is connected, [6] [2] [13] but not locally connected. [6] [13] [14] The Kirch topology is both connected and locally connected. [9] [3] [13] The integers with the Furstenberg topology form a homogeneous space, because it is a topological ring — in some sense, the only topology on for which it is a ring. [15]
It is the topology generated by the basis of all half-open intervals [a,b), where a and b are real numbers. The resulting topological space is called the Sorgenfrey line after Robert Sorgenfrey or the arrow and is sometimes written R l {\displaystyle \mathbb {R} _{l}} .
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 ...
If is a real or complex vector space and if is the set of all seminorms on then the locally convex TVS topology, denoted by , that induces on is called the finest locally convex topology on . [37] This topology may also be described as the TVS-topology on having as a neighborhood base at the origin the set of all absorbing disks in . [37] Any ...