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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.
There are many ways to define a topology on R, the set of real numbers. 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 ...
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}).
A space is completely regular if and only if every closed set can be written as the intersection of a family of zero sets in (i.e. the zero sets form a basis for the closed sets of ). A space X {\displaystyle X} is completely regular if and only if the cozero sets of X {\displaystyle X} form a basis for the topology of X . {\displaystyle X.}
The Database of Original & Non-Theoretical Uses of Topology (DONUT) is a database of scholarly articles featuring practical applications of topological data analysis to various areas of science. DONUT was started in 2017 by Barbara Giunti, Janis Lazovskis, and Bastian Rieck, [ 126 ] and as of October 2023 currently contains 447 articles. [ 127 ]
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}} .
The space of distributions, being defined as the continuous dual space of (), is then endowed with the (non-metrizable) strong dual topology induced by () and the canonical LF-topology (this topology is a generalization of the usual operator norm induced topology that is placed on the continuous dual spaces of normed spaces).
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