---
tag: 5WGMRS4JSE
kind: concept
name: Semitopological group
msc: [22A05, 54H11]
specializes: [G5S1M9]
sources:
  - url: https://en.wikipedia.org/wiki/Semitopological_group
same_as:
  wikidata: Q55632841
  wikipedia: Semitopological group
---

## Summary

A semitopological group is a [[group|G5S1M9]] with a topology in which multiplication is continuous in each variable separately.
Equivalently, every left translation and every right translation is continuous, and then each translation is a homeomorphism.
It weakens the notion of a [[topological group|92700QAB5Y]], which asks for joint continuity of multiplication and continuity of inversion.
Ellis's theorem says that a locally compact Hausdorff semitopological group is already a topological group.

## Explanations

### L1 human

In a semitopological group you may move one factor at a time and the product changes continuously.
Moving both factors at once may cause a jump.
A simple example is the group of integers under addition, with the topology in which the open sets are the empty set and the sets with finite complement.
Shifting by a fixed integer keeps such sets open, but addition of two varying integers is not continuous in this topology.