Topological group

The article on this topic in the Group Properties Wiki can be found at: topological group

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

Weaker structures

• Semitopological group
• Topological monoid
• Left-topological group
• Right-topological group

Facts

Not every topological space can be realized as the underlying space of a topological group. If we restrict attention to T0 topological groups, then the underlying space of a T0 topological group must be completely regular and homogeneous. There are other conditions that need to be satisfied. On the other hand, not every topological group is metrizable, or even normal.