# Left-topological group

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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

## Definition

A **left-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

satisfying the following equivalent conditions:

- The group multiplication map is a continuous map in terms of its
*right input*with respect to the topology. -
*Left*multiplication by any element of the group is a continuous map with respect to the topology. -
*Left*multiplication by any element of the group is a self-homeomorphism of the group with respect to the topology.

## Relation with other structures

### Stronger structures

Structure | Meaning |
---|---|

topological group | multiplication map is jointly continuous in both variables and inverse map is continuous |

paratopological group | multiplication map is jointly continuous in both variables |

quasitopological group | multiplication map is separately continuous in both variables and inverse map is continuous |

semitopological group | multiplication map is separately continuous in both variables |

### Opposite structures

- Right-topological group: The opposite group to a left-topological group, endowed with the same topology, is a right-topological group.