Topological group

Symbol-free definition
A topological group is a set endowed with the following two structures:


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

such that the following compatibility conditions are satisfied:


 * The inverse map is a continuous map from the group to itself (as a topological space map)
 * The group multiplication map is a jointly continuous map i.e. a continuous map from the Cartesian product of the group with itself, to the group (where the Cartesian product is given the product topology).

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.

Definition with symbols
A topological group is a set $$G$$ endowed with two structures:


 * The structure of a group viz a multiplication $$*$$ and an inverse map $$g \mapsto g^{-1}$$ and an identtiy element $$e$$.
 * The structure of a topological space viz a topology $$\tau$$

such that:


 * $$g \mapsto g^{-1}$$ is a continuous map with respect to $$\tau$$.
 * $$(g,h) \mapsto g * h$$ is a jointly continuous map viz it is a continuous map from $$G \times G$$ with the product topology, to $$G$$.

Weaker structures

 * Semitopological group
 * Topological monoid
 * Left-topological group
 * Right-topological group

Facts
Not every topological space can be realized as the underlying space of a topological group. If we restrict attention to T0 topological groups, then the underlying space of a T0 topological group must be completely regular and homogeneous. There are other conditions that need to be satisfied. On the other hand, not every topological group is metrizable, or even normal.