Left-topological group

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

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 group multiplication map is a left-continuous map

Definition with symbols

A topological group is a set $G$ endowed with two structures:

• The structure of a group viz a multiplication $*$ and an inverse map $g\mapsto g^{-1}$ and an identity element $e$.
• The structure of a topological space viz a topology $\tau$

such that:

• For any fixed $h$ the map $g\mapsto g*h$ is a continuous map