Editing Topological group (section)

=== Homomorphisms ===

A '''homomorphism''' of topological groups means a continuous [[group homomorphism]] {{math|''G'' → ''H''}}. 
Topological groups, together with their homomorphisms, form a [[category theory|category]]. 
A group homomorphism between topological groups is continuous if and only if it is continuous at ''some'' point.{{sfn | Narici | Beckenstein | 2011 | pp=19-45}}

An '''isomorphism''' of topological groups is a [[group isomorphism]] that is also a [[homeomorphism]] of the underlying topological spaces. 
This is stronger than simply requiring a continuous group isomorphism—the inverse must also be continuous. 
There are examples of topological groups that are isomorphic as ordinary groups but not as topological groups. 
Indeed, any non-discrete topological group is also a topological group when considered with the discrete topology. 
The underlying groups are the same, but as topological groups there is not an isomorphism.