Left-topological group

From Groupprops
Jump to: navigation, search
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:

  1. The group multiplication map is a continuous map in terms of its right input with respect to the topology.
  2. Left multiplication by any element of the group is a continuous map with respect to the topology.
  3. 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.