In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric...

functional analysis. A topological vector space is a vector space that is also a topological space with the property that the vector space operations (vector...

Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function between topological spaces that has a continuous...

mathematics, a finite topological space is a topological space for which the underlying point set is finite. That is, it is a topological space which has only...

In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a discontinuous...

Hausdorff space. More generally, all metric spaces are Hausdorff. In fact, many spaces of use in analysis, such as topological groups and topological manifolds...

is one of the principal topological properties that are used to distinguish topological spaces. A subset of a topological space X {\displaystyle X} is...

invariant under such deformations is a topological property. The following are basic examples of topological properties: the dimension, which allows...

Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces. They are generalizations of Banach spaces (normed vector spaces that...

and topological structures underlie the linear topological space (in other words, topological vector space) structure. A linear topological space is both...

mathematics, the category of topological spaces, often denoted Top, is the category whose objects are topological spaces and whose morphisms are continuous...

agree in a metric space, but may not be equivalent in other topological spaces. One such generalization is that a topological space is sequentially compact...

topology and related branches of mathematics, a topological space X is a T0 space or Kolmogorov space (named after Andrey Kolmogorov) if for every pair...

In topology and related branches of mathematics, a normal space is a topological space X that satisfies Axiom T4: every two disjoint closed sets of X have...

In mathematics, a Noetherian topological space, named for Emmy Noether, is a topological space in which closed subsets satisfy the descending chain condition...

mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space ( X , τ ) {\displaystyle (X...

of mathematics, the quotient space of a topological space under a given equivalence relation is a new topological space constructed by endowing the quotient...

In mathematics, topological groups are the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time,...

Tychonoff spaces and completely regular spaces are kinds of topological spaces. These conditions are examples of separation axioms. A Tychonoff space is any...

motivates the definitions for general topological spaces. Take X, E and a as before, but now let X be a topological space. In this case, we replace metric...

descriptions as a fallback Lindelöf space – Type of topological space Locally compact space – Type of topological space in mathematics Steen, p. 19; Willard...

T1 space is a topological space in which, for every pair of distinct points, each has a neighborhood not containing the other point. An R0 space is one...

fundamental group of a topological space is an indicator of the failure for the space to be simply connected: a path-connected topological space is simply connected...

In mathematics, a topological space is called separable if it contains a countable, dense subset; that is, there exists a sequence { x n } n = 1 ∞ {\displaystyle...

normed vector space into a metric space and a topological vector space. If this metric space is complete then the normed space is a Banach space. Every normed...

convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can...

uniform continuity and uniform convergence. Uniform spaces generalize metric spaces and topological groups, but the concept is designed to formulate the...

claimed to be homeomorphic to the topological quotient. Goreham, Anthony. Sequential convergence in Topological Spaces Archived 2011-06-04 at the Wayback...

authors use the term topologically complete for a wider class of topological spaces, the completely uniformizable spaces. A topological space homeomorphic to...

of mathematics, a topological manifold is a topological space that locally resembles real n-dimensional Euclidean space. Topological manifolds are an important...