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...

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

spaces. In 1914, Felix Hausdorff coined the term "topological space" and defined what is now called a Hausdorff space. Currently, a topological space...

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

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

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

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

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

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...

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...

and related areas of mathematics, a topological property or topological invariant is a property of a topological space that is invariant under homeomorphisms...

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

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

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

dimensional invariants of topological spaces. A topological algebra A over a topological field K is a topological vector space together with a continuous...

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

mathematics, a normal space is a topological space in which any two disjoint closed sets have disjoint open neighborhoods. Such spaces need not be Hausdorff...

In mathematics, a pointed space or based space is a topological space with a distinguished point, the basepoint. The distinguished point is just simply...

} is not a meagre topological space). A countable Hausdorff space without isolated points is meagre, whereas any topological space that contains an isolated...

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

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

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

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...

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...

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

respectively, the topological boundary, interior, and closure of S {\displaystyle S} in X {\displaystyle X} . A topological space for which there exists...

metric spaces, the greater algebraic structure of topological groups allows one to trade away some separation properties. For example, in metric spaces, a...

arbitrarily many finite discrete spaces is a Stone space, and the topological space underlying any profinite group is a Stone space. The Stone–Čech compactification...

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