---
tag: RA56GHB9JZ
kind: concept
name: Topological monoid
msc: [22A15]
specializes: [QXB6JDG68F, 0N6R902FG1]
sources:
  - url: https://en.wikipedia.org/wiki/Topological_monoid
same_as:
  wikidata: Q25098802
  wikipedia: Topological monoid
---

## Summary

A topological monoid is a [[monoid|0N6R902FG1]] with a topology in which the multiplication is jointly continuous.
It is a [[topological semigroup|QXB6JDG68F]] that has an identity element.
Topological monoids arise in algebraic topology, for example as Moore loop spaces, where loops of varying length are concatenated.
They also arise in the study of operator semigroups.

## Explanations

### L1 human

A monoid is a set with an associative operation and a neutral element, like the natural numbers under addition with neutral element $0$.
A topological monoid adds a topology so that the operation respects nearness.
An example is the set of real numbers under multiplication with the usual topology, with neutral element $1$.
It is not a group, since $0$ has no inverse, but it is a topological monoid.