Editing Topological group (section)

=== Isomorphism theorems ===

The [[isomorphism theorem]]s from ordinary group theory are not always true in the topological setting. 
This is because a bijective homomorphism need not be an isomorphism of topological groups. 

For example, a native version of the first isomorphism theorem is false for topological groups: if <math>f:G\to H</math> is a morphism of topological groups (that is, a continuous homomorphism), it is not necessarily true that the induced homomorphism <math>\tilde {f}:G/\ker f\to \mathrm{Im}(f)</math> is an isomorphism of topological groups; it will be a bijective, continuous homomorphism, but it will not necessarily be a homeomorphism. In other words, it will not necessarily admit an inverse in the [[Category theory|category]] of topological groups. For example, consider the identity map from the set of real numbers equipped with the discrete topology to the set of real numbers equipped with the Euclidean topology. This is a group homomorphism, and it is continuous because any function out of a discrete space is continuous, but it is not an isomorphism of topological groups because its inverse is not continuous.

There is a version of the first isomorphism theorem for topological groups, which may be stated as follows: if <math>f : G \to H</math> is a continuous homomorphism, then the induced homomorphism from {{math|''G''/ker(''f'')}} to {{math|im(''f'')}} is an isomorphism if and only if the map {{mvar|f}} is open onto its image.{{sfn|Bourbaki|1998|loc=section III.2.8}}

The third isomorphism theorem, however, is true more or less verbatim for topological groups, as one may easily check.