Topological semigroup
Encyclopedia
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.
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 topological semigroup is a semigroup
Semigroup
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative binary operation. A semigroup generalizes a monoid in that there might not exist an identity element...
which is simultaneously a topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
, and whose semigroup operation is continuous
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...
.
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...
is a topological semigroup.
See also
- Strongly continuous semigroup
- Analytic semigroupAnalytic semigroupIn mathematics, an analytic semigroup is particular kind of strongly continuous semigroup. Analytic semigroups are used in the solution of partial differential equations; compared to strongly continuous semigroups, analytic semigroups provide better regularity of solutions to initial value...
- Ellis–Nakamura lemmaEllis–Nakamura lemmaIn mathematics, the Ellis–Numakura lemma states that if S is a non-empty semigroup with a topology such thatS is compact and the product is semi-continuous, then S has an idempotent element p, .-Applications:...