Left-topological group
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This article gives a basic definition in the following area: topological group theory
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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.
