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This is a list of useful examples in general topology, a field of mathematics. Alexandrov topology; Cantor space; Co-kappa topology Cocountable topology; Cofinite topology; Compact-open topology; Compactification; Discrete topology; Double-pointed cofinite topology; Extended real number line; Finite topological space; Hawaiian earring; Hilbert cube
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance.More specifically, a topological space is a set whose elements are called points, along with an additional structure called a topology, which can be defined as a set of neighbourhoods for each point that satisfy some axioms ...
A topological space is a set endowed with a structure, called a topology, which allows defining continuous deformation of subspaces, and, more generally, all kinds of continuity. Euclidean spaces , and, more generally, metric spaces are examples of a topological space, as any distance or metric defines a topology.
The following is a list of named topologies or topological spaces, many of which are counterexamples in topology and related branches of mathematics. This is not a list of properties that a topology or topological space might possess; for that, see List of general topology topics and Topological property.
A manifold is a topological space that near each point resembles Euclidean space. Examples include the plane , the sphere , and the torus , which can all be realized in three dimensions, but also the Klein bottle and real projective plane which cannot be embedded in three dimensions, but can be embedded in four dimensions.
Cardinal functions are widely used in topology as a tool for describing various topological properties. [4] [5] Below are some examples.(Note: some authors, arguing that "there are no finite cardinal numbers in general topology", [6] prefer to define the cardinal functions listed below so that they never take on finite cardinal numbers as values; this requires modifying some of the definitions ...
The term symmetric space also has another meaning.) A topological space is a T 1 space if and only if it is both an R 0 space and a Kolmogorov (or T 0) space (i.e., a space in which distinct points are topologically distinguishable). A topological space is an R 0 space if and only if its Kolmogorov quotient is a T 1 space.
A closed subspace of a Baire space need not be Baire. See the Examples section. If a space contains a dense subspace that is Baire, it is also a Baire space. [17] A space that is locally Baire, in the sense that each point has a neighborhood that is a Baire space, is a Baire space. [18] [19] Every topological sum of Baire spaces is Baire. [20]