---
tag: H0V3TNH7DN
kind: concept
name: Paratopological group
msc: [22A05, 54H11]
specializes: [RA56GHB9JZ]
sources:
  - url: https://en.wikipedia.org/wiki/Paratopological_group
same_as:
  wikidata: Q7136179
  wikipedia: Paratopological group
---

## Summary

A paratopological group is a [[group|G5S1M9]] with a topology in which multiplication is jointly continuous, while inversion need not be continuous.
It sits between a [[semitopological group|5WGMRS4JSE]] and a [[topological group|92700QAB5Y]].
The standard example that is not a topological group is the real line under addition with the lower limit topology.
Much of the theory asks which extra conditions force inversion to be continuous.

## Explanations

### L1 human

In a paratopological group, multiplying nearby elements gives nearby results, but taking inverses may tear the space apart.
Picture the real line where the basic open sets are the half-open intervals $[a, b)$.
Adding two numbers still behaves well: if $x$ and $y$ move a little to the right, so does $x + y$.
But the map $x \mapsto -x$ turns intervals $[a, b)$ into intervals $(-b, -a]$, which are not open in this topology.