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. [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 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
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. [2]
A very special kind of topological extensions are the ones of the form where is the unit circle and and are topological abelian groups. [3]
A topological abelian group belongs to the class if and only if every topological extension of the form splits
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. [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 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
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. [2]
A very special kind of topological extensions are the ones of the form where is the unit circle and and are topological abelian groups. [3]
A topological abelian group belongs to the class if and only if every topological extension of the form splits