---
tag: ZZK2GM1V15
kind: concept
name: Continuous multiplication
msc: [22A15, 22A05]
sources:
  - url: https://en.wikipedia.org/wiki/Topological_group
---

## Summary

A set with a binary operation and a topology has continuous multiplication when the operation, seen as a map of two variables, is continuous.
Here the domain carries the [[product topology|0GBG8B49AA]].
This single condition is the common base of [[topological semigroups|QXB6JDG68F]], [[topological monoids|RA56GHB9JZ]] and [[topological groups|92700QAB5Y]].
It asks for joint continuity, which is stronger than continuity in each variable separately.

## Explanations

### L1 human

Think of multiplying two numbers $x$ and $y$.
If you move $x$ and $y$ a little, the product $xy$ also moves only a little.
That is what continuous multiplication means: small changes in both inputs give a small change in the output.
For example, ordinary multiplication on the real line has this property, so $(\R, \cdot)$ has continuous multiplication.