# Topological group

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

## Contents

## 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 , 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 -topological group. `Further information: T0 topological group`

### Definition with symbols

A **topological group** is a set endowed with two structures:

- The structure of a group viz a multiplication and an inverse map and an identtiy element .
- The structure of a topological space viz a topology

such that:

- is a continuous map with respect to .
- is a jointly continuous map viz it is a continuous map from with the product topology, to .

## Relation with other structures

### Weaker structures

## 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.

`Further information: Topological group not implies normal`