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In mathematics, a topological semigroup is a semigroup that is simultaneously a topological space, and whose semigroup operation is continuous.^[1]

Every topological group is a topological semigroup.

Analytic semigroup – Type of strongly continuous semigroup
Compact group – Topological group with compact topology
Complete field – algebraic structure that is complete relative to a metric
Ellis–Numakura lemma – A compact topological semigroup with a semicontinuous product has an idempotent element
Locally compact group – topological group for which the underlying topology is locally compact and Hausdorff, so that the Haar measure can be defined
Locally compact quantum group – relatively new C*-algebraic approach toward quantum groups
Ordered topological vector space
Strongly continuous semigroup – Generalization of the exponential function
Topological abelian group – topological group whose group is abelian
Topological field – Algebraic structure with addition, multiplication, and division
Topological group – Group that is a topological space with continuous group action
Topological module
Topological ring – ring where ring operations are continuous
Topological vector lattice
Topological vector space – Vector space with a notion of nearness

References

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