mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a paratopological group is a topological semigroup that is algebraically a

group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic iden ...

. In other words, it is a group ''G'' with a topology such that the group's product operation is a

continuous function In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as '' discontinuities''. More preci ...

from ''G'' × ''G'' to ''G''. This differs from the definition of a

topological group In mathematics, topological groups are the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two structures ...

in that the group inverse is not required to be continuous. As with topological groups, some authors require the topology to be Hausdorff.A. V. Arhangelskii
Topological spaces connected to algebraic structures
Compact paratopological groups are automatically topological groups.

References

Topological groups {{topology-stub