Quasitopological group

Abstract definition
A quasitopological group is a group equipped with a topology such that all the defining operations of a group (in the universal algebra sense) are separately continuous, i.e., each of the defining operations is continuous separately in each of its inputs.

Concrete definition
A quasitopological group is a set endowed with the following two structures:


 * The structure of a group, viz., an associative binary operation with identity element and inverses
 * The structure of a topological space

satisfying the following compatibility conditions:

Due to the fact that inverse map is involutive and hence that every group is naturally isomorphic to its opposite group via the inverse map, the continuity of the inverse map implies that simply assuming continuity of the multiplication in any one of its inputs implies separate continuity of multiplication. In particular, any left topological group where the inverse map is continuous is a quasitopological group, and similarly, any right topological group where the inverse map is continuous is a quasitopological group.

Incomparable structures

 * Paratopological group: Here, the group multiplication is required to be jointly continuous, but we make no assumption about the continuity of the inverse map.