Editing Topological group (section)

== Formal definition ==

A '''topological group''', {{mvar|G}}, is a [[topological space]] that is also a group such that the group operation (in this case product):
:{{math|⋅ : ''G'' × ''G'' → ''G''}},  {{math|    (''x'', ''y'') ↦ ''xy''}}
and the inversion map:
:{{math|<sup>−1</sup> : ''G'' → ''G''}},  {{math| ''x'' ↦ ''x''<sup>−1</sup>}}
are [[Continuous function|continuous]].<ref group=note>''i.e.'' Continuous means that for any open set {{math|''U'' ⊆ ''G''}}, {{math|''f''<sup>−1</sup>(''U'')}} is open in the domain {{math|dom ''f''}} of {{mvar|f}}.</ref> 
Here {{math|''G'' × ''G''}} is viewed as a topological space with the [[product topology]]. 
Such a topology is said to be '''compatible with the group operations''' and is called a '''group topology'''.

;Checking continuity

The product map is continuous if and only if for any {{math|''x'', ''y'' ∈ ''G''}} and any neighborhood {{mvar|W}} of {{math|''xy''}} in {{mvar|G}}, there exist neighborhoods {{mvar|U}} of {{mvar|x}} and {{mvar|V}} of {{mvar|y}} in {{mvar|G}} such that {{math|''U'' ⋅ ''V'' ⊆ ''W''}}, where {{math|''U'' ⋅ ''V'' :{{=}} {''u'' ⋅ ''v'' : ''u'' ∈ ''U'', ''v'' ∈ ''V''}}}. 
The inversion map is continuous if and only if for any {{math|''x'' ∈ ''G''}} and any neighborhood {{mvar|V}} of {{math|''x''<sup>−1</sup>}} in {{mvar|G}}, there exists a neighborhood {{mvar|U}} of {{mvar|x}} in {{mvar|G}} such that {{math|''U''<sup>−1</sup> ⊆ ''V''}}, where {{math|''U''<sup>−1</sup> :{{=}} { ''u''<sup>−1</sup> : ''u'' ∈ ''U'' }}}.

To show that a topology is compatible with the group operations, it suffices to check that the map
:{{math|''G'' × ''G'' → ''G''}}, {{math|(''x'', ''y'') ↦ ''xy''<sup>−1</sup>}}
is continuous. 
Explicitly, this means that for any {{math|''x'', ''y'' ∈ ''G''}} and any neighborhood {{mvar|W}} in {{mvar|G}} of {{math|''xy''<sup>−1</sup>}}, there exist neighborhoods {{mvar|U}} of {{mvar|x}} and {{mvar|V}} of {{mvar|y}} in {{mvar|G}} such that {{math|''U'' ⋅ (''V''<sup>−1</sup>) ⊆ ''W''}}.

;Additive notation

This definition used notation for multiplicative groups; 
the equivalent for additive groups would be that the following two operations are continuous: 
:{{math|+ : ''G'' × ''G'' → ''G ''}}, {{math|  (''x'', ''y'') ↦ ''x'' + ''y''}}
:{{math|− : ''G'' → ''G ''}}, {{math|''x'' ↦ −''x''}}.

;Hausdorffness

Although not part of this definition, many authors<ref>{{harvnb|Armstrong|1997|p=73}}; {{harvnb|Bredon|1997|p=51}}</ref> require that the topology on {{mvar|G}} be [[Hausdorff space|Hausdorff]]. 
One reason for this is that any topological group can be canonically associated with a Hausdorff topological group by taking an appropriate canonical quotient; 
this however, often still requires working with the original non-Hausdorff topological group.
Other reasons, and some equivalent conditions, are discussed below.

This article will not assume that topological groups are necessarily Hausdorff.

;Category

In the language of [[category theory]], topological groups can be defined concisely as [[group object]]s in the [[category of topological spaces]], in the same way that ordinary groups are group objects in the [[category of sets]]. 
Note that the axioms are given in terms of the maps (binary product, unary inverse, and nullary identity), hence are categorical definitions.

=== 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.