Topological group

Abstract definition
The notion of topological group can be defined in the following equivalent ways:


 * 1) In the language of universal algebra, it is a group equipped with a topology for which all the defining operations of groups are continuous.
 * 2) It is a group object in the category of topological spaces.

Concrete definition
A topological group is a set $$G$$ endowed with the following two structures:


 * The structure of a group, i.e., a binary operation called multiplication or product, a constant called the identity element, and a unary operation called the inverse map, and satisfying the conditions for a group
 * The structure of a topological space

such that the following compatibility conditions are satisfied:

T0 assumption
Some people assume a topological group to be $$T_0$$, that is, that there is no pair of points with each in the closure of the other. This is not a very restrictive assumption, because if we quotient out a topological group by the closure of the identity element, we do get a $$T_0$$-topological group. However, the definition above does not include this assumption.

Caution about algebraic groups
Algebraic groups can be given a topology arising from their algebraic variety structure, namely, the Zariski topology. However, with some trivial exceptions, algebraic groups are not topological groups. The reason is that the multiplication map is not jointly continuous. Another way of seeing this is that algebraic groups are $$T_1$$ but not Hausdorff under the Zariski topology, but we know that for topological groups, being $$T_1$$ is equivalent to being Hausdorff.

However, there are two saving graces:


 * Algebraic groups are quasitopological groups, where a quasitopological group is a group with a topology where the inverse map is continuous and the group multiplication map is separately continuous in each variable.
 * Algebraic groups over a topological field are topological groups with respect to the topology arising from the field topology, and in the case that the field comes equipped with analytic structure, they become Lie groups over the field. For instance, real algebraic groups are real Lie groups and complex algebraic groups are complex Lie groups.

Stronger structures

 * Lie group

Textbook references

 * , Page 145, Supplementary Exercises (assumes $$T_0$$ in the definition of topological group)