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 formula_1 where formula_2 and formula_3 are topological groups and formula_4 and formula_5 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 (or congruent) if there exists a topological isomorphism formula_8 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 formula_11 is the natural inclusion over the first factor and formula_12 is the natural projection over the second factor.
It is easy to prove that the topological extension formula_6 splits if and only if there is a continuous homomorphism formula_14 such that formula_15 is the identity map on formula_16
Note that the topological extension formula_6 splits if and only if the subgroup formula_18 is a topological direct summand of formula_19
Extensions of locally compact abelian groups (LCA).
An extension of topological abelian groups will be a short exact sequence formula_1 where formula_2 and formula_3 are locally compact abelian groups and formula_4 and formula_5 are relatively open continuous homomorphisms.