Paratopological group
Encyclopedia
In mathematics
, a paratopological group is a topological semigroup
that is algebraically a group
. In other words, it is a group G with a topology such that the group's product operation is a continuous function
from G × G to G. This differs from the definition of a topological group
in that the group inverse is not required to be continuous.
As with topological groups, some authors require the topology to be Hausdorff
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a paratopological group is a topological semigroup
Topological semigroup
In mathematics, a topological semigroup is a semigroup which is simultaneously a topological space, and whose semigroup operation is continuous.A topological group is a topological semigroup.- See also :* Strongly continuous semigroup* Analytic semigroup...
that is algebraically a group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
. In other words, it is a group G with a topology such that the group's product operation is a continuous function
Continuous function
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
from G × G to G. This differs from the definition of a topological group
Topological group
In mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the group's inverse function are continuous functions with respect to the topology. A topological group is a mathematical object with both an algebraic structure and a...
in that the group inverse is not required to be continuous.
As with topological groups, some authors require the topology to be Hausdorff
Hausdorff space
In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently...
.