mathematics, a topological module is a module over a topological ring such that scalar multiplication and addition are continuous. A topological vector space...

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...

mathematics, a topological abelian group, or TAG, is a topological group that is also an abelian group. That is, a TAG is both a group and a topological space...

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

In applied mathematics, topological data analysis (TDA) is an approach to the analysis of datasets using techniques from topology. Extraction of information...

be generalized to topological modules. A subset A {\displaystyle A} of a topological module M {\displaystyle M} over a topological ring R {\displaystyle...

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

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

of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize...

homology theories for topological spaces that produce the same answer, one also often speaks of the homology of a topological space. (This latter notion...

G} -module (it is an abelian group under addition). If G {\displaystyle G} is a topological group and M {\displaystyle M} is an abelian topological group...

to the topology, the module operations of addition and scalar multiplication are continuous, so that M becomes a topological module. However, M need not...

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

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

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

aforementioned block form. A topological vector space (TVS) X , {\displaystyle X,} such as a Banach space, is said to be a topological direct sum of two vector...

persistence module is a mathematical structure in persistent homology and topological data analysis that formally captures the persistence of topological features...

This is a list of general topology topics. Topological space Topological property Open set, closed set Clopen set Closure Boundary Density G-delta set...

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

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

{\displaystyle X} into a topological vector space). Every topological vector space (TVS) X {\displaystyle X} is an additive commutative topological group but not...

any of several related functors on rings and modules that result in complete topological rings and modules. Completion is similar to localization, and...

It is in this form that the regular representation is generalized to topological groups such as Lie groups. The specific definition in terms of W is as...

the prime ideals of a commutative ring a topological space equipped with the Zariski topology; this topological space is called the spectrum of the ring...

convex topological vector space – Vector space with a topology defined by convex open sets Metrizable topological vector space – A topological vector...

mathematical physics, a topological quantum field theory (or topological field theory or TQFT) is a quantum field theory that computes topological invariants. While...

In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks...

is a topological space where distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space,...

group G in an associated G-module M to elucidate the properties of the group. By treating the G-module as a kind of topological space with elements of G...