# Paratopological group

## Contents

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 paratopological group is a set $G$ endowed with the following two structures:

• The structure of a group, viz., an associative binary operation with identity element and inverses
• The structure of a topological space

such that the following are true:

Multiplication or product 2 (so it's a map $G \times G \to G$) continuous as a map from $G \times G$ (equipped with the product topology) to $G$. In other words, the group multiplication $(g,h) \mapsto gh$ is jointly continuous. Joint continuity is strictly stronger than separate continuity, which would mean continuity in each input holding the other input fixed.
Identity element 0 (it's a constant element $e \in G$) no condition. As such, we may impose the condition that the map from a one-point space to $G$ sending the point to the identity element is continuous, but this condition is vacuously true.
Inverse map 1 (so it's a map $G \to G$) no condition. In particular, we do not require this operation to be continuous. If we did impose the additional condition that the inverse map be continuous, we would get the definition of topological group.

## Relation with other structures

### Stronger structures

Structure Meaning Proof of implication Proof of strictness (reverse implication failure)
topological group group multiplication is jointly continuous and inverse map is continuous (obvious) paratopological group not implies topological group

### Weaker structures

Structure Meaning
semitopological group group multiplication is separately continuous in both variables
left-topological group group multiplication is continuous in right input, i.e., left multiplication maps are continuous
right-topological group group multiplication is continuous in left input, i.e., right multiplication maps are continuous

### Incomparable structures

• Quasitopological group means the group multiplication is separately continuous but the inverse map is required to be continuous.