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 ${\displaystyle 0\to H{\stackrel {\imath }{\to }}X{\stackrel {\pi }{\to }}G\to 0}$ where ${\displaystyle H,X}$ and ${\displaystyle G}$ are topological groups and ${\displaystyle i}$ and ${\displaystyle \pi }$ are continuous homomorphisms which are also open onto their images.^[1] Every extension of topological groups is therefore a group extension.