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In mathematics, more specifically in topological groups, an extension of topological groups, or a topological extension, is a short exact sequence 0 → H → ı X → π G → 0 {\displaystyle 0\to H{\stackrel {\imath }{\to }}X{\stackrel {\pi }{\to }}G\to 0} where H , X {\displaystyle H,X} and G {\displaystyle G} are topological groups and i {\displaystyle i} and π {\displaystyle \pi } are continuous homomorphisms which are also open onto their images. Every extension of topological groups is therefore a group extension.