---
tag: T3A3JTW6Q4
kind: concept
name: Isomorphism of topological groups
msc: [22A05]
specializes: [9J5PQMG3HJ]
sources:
  - url: https://en.wikipedia.org/wiki/Topological_group
---

## Summary

An isomorphism of topological groups is a group isomorphism that is also a homeomorphism.
Two isomorphic topological groups are the same in every respect that involves both the group law and the topology.
A continuous bijective homomorphism need not be an isomorphism, because its inverse may fail to be continuous.
The open mapping theorem for locally compact groups gives conditions under which this cannot happen.

## Explanations

### L1 human

Two topological groups are isomorphic when one can rename the elements of one to get the other, keeping both the multiplication and the notion of nearness.
For example, the positive real numbers under multiplication and the real numbers under addition are isomorphic.
The logarithm turns products into sums, and both the logarithm and the exponential function are continuous.
So these two groups are copies of each other, even though they look different at first.