Definition 1

A topological group is a set G which carries a group structure and a topology and satisfies the following two axioms:

(GT_I). The mapping $(x, y) \to xy$ of $G \times G$ into $G$ is continuous.

(GT_{II}). The mapping $x \to x^{-1}$ of $G$ into $G$ (the symmetry of the group G) is continuous.

A group structure and a topology on a set G are said to be compatible if they satisfy (GT_I) and (GT_{II}).