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

are topological groups and

are continuous homomorphisms which are also open onto their images.

[1] Every extension of topological groups is therefore a group extension.

We say that the topological extensions and are equivalent (or congruent) if there exists a topological isomorphism

making commutative the diagram of Figure 1.

We say that the topological extension is a split extension (or splits) 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

is the identity map on

Note that the topological extension

splits if and only if the subgroup

is a topological direct summand of

An extension of topological abelian groups will be a short exact sequence

are locally compact abelian groups and

are relatively open continuous homomorphisms.

[2] A very special kind of topological extensions are the ones of the form

is the unit circle and

are topological abelian groups.

[3] A topological abelian group

belongs to the class

if and only if every topological extension of the form