Topological group

This article gives a basic definition in the following area: topological group theory
View other basic definitions in topological group theory |View terms related to topological group theory |View facts related to topological group theory

This article describes a compatible combination of two structures: group and topological space

This article defines the notion of group object in the category of topological spaces|View other types of group objects

Definition

Symbol-free definition

A topological group is a set endowed with the following two structures:

• The structure of a group, viz a binary operation called multiplication or product, a unary operation called the inverse map, and a constant called the identity element satisfying the conditions for a group
• The structure of a topological space

such that the following compatibility conditions are satisfied:

• The inverse map is a continuous map from the group to itself (as a topological space map)
• The group multiplication map is a jointly continuous map i.e. a continuous map from the Cartesian product of the group with itself, to the group (where the Cartesian product is given the product topology).

Some people assume a topological group to be $T_{0}$, that is, that there is no pair of points with each in the closure of the other. This is not a very restrictive assumption, because if we quotient out a topological group by the closure of the identity element, we do get a $T_{0}$-topological group. Further information: T0 topological group

Definition with symbols

A topological group is a set $G$ endowed with two structures:

• The structure of a group viz a multiplication $*$ and an inverse map $g\mapsto g^{-1}$ and an identtiy element $e$.
• The structure of a topological space viz a topology $\tau$

such that:

• $g\mapsto g^{-1}$ is a continuous map with respect to $\tau$.
• $(g,h)\mapsto g*h$ is a jointly continuous map viz it is a continuous map from $G\times G$ with the product topology, to $G$.

Relation with other structures

Stronger structures

• Algebraic group over a topological field
• Lie group

Weaker structures

• Semitopological group
• Left-topological group
• Right-topological group

References

Textbook references

• Topology (2nd edition) by James R. Munkres^{More info}, Page 145, Supplementary Exercises (assumes $T_{0}$ in the definition of topological group)