Extension of a topological group


In mathematics, more specifically in topological groups, an extension of topological groups, or a topological extension, is a short exact sequence where and are topological groups and and are continuous homomorphisms which are also open onto their images. Every extension of topological groups is therefore a group extension.

Classification of extensions of topological groups

We say that the topological extensions
and
are equivalent if there exists a topological isomorphism making commutative the diagram of Figure 1.
We say that the topological extension
is a split extension if it is equivalent to the trivial extension
where is the natural inclusion over the first factor and is the natural projection over the second factor.
It is easy to prove that the topological extension splits if and only if there is a continuous homomorphism such that is the identity map on
Note that the topological extension splits if and only if the subgroup is a topological direct summand of

Examples

An extension of topological abelian groups will be a short exact sequence where and are locally compact abelian groups and and are relatively open continuous homomorphisms.