Left-topological group

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.

Opposite structures

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