---
tag: QXB6JDG68F
kind: concept
name: Topological semigroup
msc: [22A15]
specializes: [0XMEM406JM]
sources:
  - url: https://en.wikipedia.org/wiki/Topological_semigroup
same_as:
  wikidata: Q7825043
  wikipedia: Topological semigroup
---

## Summary

A topological semigroup is a [[semigroup|0XMEM406JM]] with a topology in which the semigroup operation is jointly continuous.
It is a very general setting where algebra and topology meet through one associative operation.
Topological monoids and topological groups are special cases.
Compact topological semigroups have a rich structure theory with uses in topological dynamics.

## Explanations

### L1 human

A semigroup is a set with an associative way of combining two elements, without any demand for an identity or inverses.
A topological semigroup adds a notion of nearness, and asks that combining nearby elements gives nearby results.
For example, the interval $[0, 1]$ with ordinary multiplication is a topological semigroup.
The product of two numbers in $[0, 1]$ is again in $[0, 1]$, and the product depends continuously on both factors.