---
tag: PRQKKB7HNX
kind: definition
name: Paratopological group
msc: [22A05, 54H11]
defines: H0V3TNH7DN
uses: [G5S1M9, 0CQHFPGJ45, ZZK2GM1V15, 92700QAB5Y, Y4T3GVK8GV]
sources:
  - url: https://en.wikipedia.org/wiki/Paratopological_group
---

## Definition

A *paratopological group* is a [[group|G5S1M9]] $G$ together with a [[topology|0CQHFPGJ45]] on $G$ such that the multiplication $G \times G \to G$, $(x, y) \mapsto xy$, is a [[continuous multiplication|ZZK2GM1V15]].
No condition is placed on the inversion $x \mapsto x^{-1}$.

## Remarks

Example: $(\R, +)$ with the topology of the [[Sorgenfrey line|Y4T3GVK8GV]], generated by the intervals $[a, b)$.
If $x + y \in [c, d)$, choose $\varepsilon > 0$ with $x + y + 2\varepsilon < d$; then $[x, x + \varepsilon) + [y, y + \varepsilon) \subseteq [x + y, x + y + 2\varepsilon) \subseteq [c, d)$, so addition is continuous.
Inversion is not continuous, since the preimage of $[0, 1)$ under $x \mapsto -x$ is $(-1, 0]$, which is not open.
So this is a paratopological group that is not a [[topological group|92700QAB5Y]].
Non-example: $(\Z, +)$ with the cofinite topology is a group, but addition is not jointly continuous, because every nonempty open set $U \times V$ of $\Z \times \Z$ contains a pair $(x, -x)$.
A paratopological group is the same as a topological monoid whose underlying monoid is a group.