example, in physics. In functional analysis, every topological vector space is an additive topological group with the additional property that scalar multiplication...

abelian groups Protorus – Mathematical object, a topological abelian group that is compact and connected Ordered topological vector space Topological field –...

In mathematics, a topological ring is a ring R {\displaystyle R} that is also a topological space such that both the addition and the multiplication are...

considerations. A topological group is a locally compact group if the underlying topological space is locally compact and Hausdorff; a topological group is abelian...

In mathematics, a topological group G is called a discrete group if there is no limit point in it (i.e., for each element in G, there is a neighborhood...

redirect targets Topological group – Group that is a topological space with continuous group action Topological module Topological ring – ring where...

mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops...

a compact complete set that is not closed. Any topological vector space is an abelian topological group under addition, so the above conditions apply....

onto their images. Every extension of topological groups is therefore a group extension. We say that the topological extensions 0 → H → i X → π G → 0 {\displaystyle...

topological spaces, generally exemplary of automorphism groups and topologically invariant in the group isomorphism sense. There is a natural group action...

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

addition to continuous actions of topological groups on topological spaces, one also often considers smooth actions of Lie groups on smooth manifolds, regular...

compact (topological) group is a topological group whose topology realizes it as a compact topological space (when an element of the group is operated...

covering group of a topological group H is a covering space G of H such that G is a topological group and the covering map p : G → H is a continuous group homomorphism...

representations of the group. As such, they are similar to the group ring associated to a discrete group. If G is a locally compact Hausdorff group, G carries an...

Such groups are called topological groups, and they are the group objects in the category of topological spaces. The most basic examples are the group of...

is a topological space and Y is an additive topological group (i.e. a group endowed with a topology making its operations continuous). Topological vector...

infinite abstract groups and topological groups: whenever a group Γ can be realized as a lattice in a topological group G, the geometry and analysis pertaining...

satisfies the above topological definition. Conversely, let G be a topological group that is a Lie group in the above topological sense and choose an...

topological group Local (disambiguation) Localization (disambiguation) This disambiguation page lists articles associated with the title Local Group....

In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures...

well be stated in the context of a topological group: A sequence ( x k ) {\displaystyle (x_{k})} in a topological group G {\displaystyle G} is a Cauchy sequence...

GL(n, K), the group of n × n {\displaystyle n\times n} invertible matrices on the field K. If G is a topological group and V is a topological vector space...

example of a group object is a topological group, a group whose underlying set is a topological space such that the group operations are continuous. Formally...

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

Manfred P. (1999). Topological Vector Spaces (2nd ed.). New York: Springer. ISBN 978-1-4612-7155-0. Trèves, François (1967). Topological Vector Spaces, Distributions...

Common types of topological spaces include Euclidean spaces, metric spaces and manifolds. Although very general, the concept of topological spaces is fundamental...

profinite group is a topological group that is in a certain sense assembled from a system of finite groups. The idea of using a profinite group is to provide...

In mathematics, a compactly generated (topological) group is a topological group G which is algebraically generated by one of its compact subsets. This...

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