Topological groups, Lie groups Definition

Isomorphism of topological groups

Builds on Topological group Continuous group homomorphism Group isomorphism Homeomorphism

Definition

Let $G$ and $H$ be topological groups. An isomorphism of topological groups from $G$ to $H$ is a map $f \colon G \to H$ that is both a group isomorphism and a homeomorphism. Equivalently, $f$ is a bijective continuous group homomorphism whose inverse $f^{-1} \colon H \to G$ is continuous. The groups $G$ and $H$ are isomorphic if such a map exists.

Remarks

Example: $\log \colon (\,(0, \infty), \cdot\,) \to (\R, +)$ . It is bijective, $\log(xy) = \log x + \log y$ , and both $\log$ and its inverse $\exp$ are continuous. Non-example: the identity map from $(\R, +)$ with the discrete topology to $(\R, +)$ with the usual topology. It is a continuous bijective homomorphism, but its inverse is not continuous, since $\set{0}$ is open in the discrete topology and not in the usual one.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Continuous group homomorphism concept
Group isomorphism concept
Homeomorphism concept
Topological group concept
Isomorphism of topological groups definition

Tag	`HEKY61PKA9`
URI	`m2://HEKY61PKA9`
Kind	definition
Writing	W0 stub
MSC	22A05
Defines	Isomorphism of topological groups
Uses	Topological group, Continuous group homomorphism, Group isomorphism, Homeomorphism

Isomorphism of topological groups

Definition

Remarks

Reading path

Further reading

What this page needs for W1

What links here