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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.

In mathematics, more specifically in general topology, **compactness** is a property that generalizes the notion of a subset of Euclidean space being closed and bounded. Examples include a closed interval, a rectangle, or a finite set of points. This notion is defined for more general topological spaces than Euclidean space in various ways.

In mathematics, a **continuous function** is a function for which sufficiently small changes in the input result in arbitrarily small changes in the output. Otherwise, a function is said to be a *discontinuous* function. A continuous function with a continuous inverse function is called a homeomorphism.

In mathematics, a **topological vector space** is one of the basic structures investigated in functional analysis. A topological vector space is a vector space which is also a topological space, the latter thereby admitting a notion of continuity. More specifically, its topological space has a uniform topological structure, allowing a notion of uniform convergence.

In mathematics, a **topological ring** is a ring *R* that is also a topological space such that both the addition and the multiplication are continuous as maps

In mathematics, **general topology** is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology. Another name for general topology is **point-set topology**.

In functional analysis and related branches of mathematics, the **Banach–Alaoglu theorem** states that the closed unit ball of the dual space of a normed vector space is compact in the weak* topology. A common proof identifies the unit ball with the weak* topology as a closed subset of a product of compact sets with the product topology. As a consequence of Tychonoff's theorem, this product, and hence the unit ball within, is compact.

In mathematics, a function between topological spaces is called **proper** if inverse images of compact subsets are compact. In algebraic geometry, the analogous concept is called a proper morphism.

In mathematics, a ** C_{0}-semigroup**, also known as a

In functional analysis and related areas of mathematics, **barrelled spaces** are Hausdorff topological vector spaces for which every barrelled set in the space is a neighbourhood for the zero vector. A **barrelled set** or a **barrel** in a topological vector space is a set which is convex, balanced, absorbing and closed. Barrelled spaces are studied because a form of the Banach–Steinhaus theorem still holds for them.

In mathematics, particularly in functional analysis, a **bornological space** is a type of space which, in some sense, possesses the minimum amount of structure needed to address questions of boundedness of sets and functions, in the same way that a topological space possesses the minimum amount of structure needed to address questions of continuity. Bornological spaces were first studied by Mackey. The name was coined by Bourbaki after *borné*, the French word for "bounded".

In topology and related fields of mathematics, a **sequential space** is a topological space that satisfies a very weak axiom of countability. Sequential spaces are the most general class of spaces for which sequences suffice to determine the topology.

In mathematics, a **representation theorem** is a theorem that states that every abstract structure with certain properties is isomorphic to another structure.

In mathematics, in the field of topology, a topological space is said to be **pseudocompact** if its image under any continuous function to **R** is bounded. Many authors include the requirement that the space be completely regular right in the definition of pseudocompactness. Pseudocompact spaces were defined by Edwin Hewitt in 1948.

In mathematics, the **Krylov–Bogolyubov theorem** may refer to either of the two related fundamental theorems within the theory of dynamical systems. The theorems guarantee the existence of invariant measures for certain "nice" maps defined on "nice" spaces and were named after Russian-Ukrainian mathematicians and theoretical physicists Nikolay Krylov and Nikolay Bogolyubov who proved the theorems.

In mathematics, a topological space *X* is **sequentially compact** if every sequence of points in *X* has a convergent subsequence converging to a point in *X*.

In mathematics, a topological space *X* is said to be **limit point compact** or **weakly countably compact** if every infinite subset of *X* has a limit point in *X*. This property generalizes a property of compact spaces. In a metric space, limit point compactness, compactness, and sequential compactness are all equivalent. For general topological spaces, however, these three notions of compactness are not equivalent.

In mathematics, **topological dynamics** is a branch of the theory of dynamical systems in which qualitative, asymptotic properties of dynamical systems are studied from the viewpoint of general topology.

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.

**Semigroup Forum** is a mathematics research journal published by Springer. The journal serves as a platform for the speedy and efficient transmission of information on current research in semigroup theory. Coverage in the journal includes: algebraic semigroups, topological semigroups, partially ordered semigroups, semigroups of measures and harmonic analysis on semigroups, transformation semigroups, and applications of semigroup theory to other disciplines such as ring theory, category theory, automata, and logic. Semigroups of operators were initially considered off-topic, but began being included in the journal in 1985.

In mathematics, and more precisely in semigroup theory, a **variety of finite semigroups** is a class of semigroups having some nice algebraic properties. Those classes can be defined in two distinct way, using either algebraic notions or topological notions. Varieties of finite monoids, varieties of finite ordered semigroups and varieties of finite ordered monoids are defined similarly.

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