# Topological Boolean algebra

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Topological Boolean algebra may refer to:

• In abstract algebra and mathematical logic, topological Boolean algebra is one of the many names that have been used for an interior algebra in the literature.
• In the work of the mathematician R.S. Pierce, a topological Boolean algebra is a Boolean algebra equipped with both a closure operator and a derivative operator generalizing T1 topological spaces and may be considered to be a special case of interior algebras rather than synonymous with them.
• In topological algebra — the study of topological spaces with algebraic structure, a topological Boolean algebra is a Boolean algebra endowed with a topological structure in which the operations of complement, join, and meet are continuous functions.

In algebra, which is a broad division of mathematics, abstract algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras. The term abstract algebra was coined in the early 20th century to distinguish this area of study from the other parts of algebra.

Mathematical logic is a subfield of mathematics exploring the applications of formal logic to mathematics. It bears close connections to metamathematics, the foundations of mathematics, and theoretical computer science. The unifying themes in mathematical logic include the study of the expressive power of formal systems and the deductive power of formal proof systems.

In abstract algebra, an interior algebra is a certain type of algebraic structure that encodes the idea of the topological interior of a set. Interior algebras are to topology and the modal logic S4 what Boolean algebras are to set theory and ordinary propositional logic. Interior algebras form a variety of modal algebras.

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In abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice. This type of algebraic structure captures essential properties of both set operations and logic operations. A Boolean algebra can be seen as a generalization of a power set algebra or a field of sets, or its elements can be viewed as generalized truth values. It is also a special case of a De Morgan algebra and a Kleene algebra.

In mathematics, and more specifically in abstract algebra, an algebraic structure on a set A is a collection of finitary operations on A; the set A with this structure is also called an algebra.

In topology, a clopen set in a topological space is a set which is both open and closed. That this is possible may seem counter-intuitive, as the common meanings of open and closed are antonyms, but their mathematical definitions are not mutually exclusive. A set is closed if its complement is open, which leaves the possibility of an open set whose complement is also open, making both sets both open and closed, and therefore clopen.

Noncommutative geometry (NCG) is a branch of mathematics concerned with a geometric approach to noncommutative algebras, and with the construction of spaces that are locally presented by noncommutative algebras of functions. A noncommutative algebra is an associative algebra in which the multiplication is not commutative, that is, for which does not always equal ; or more generally an algebraic structure in which one of the principal binary operations is not commutative; one also allows additional structures, e.g. topology or norm, to be possibly carried by the noncommutative algebra of functions.

In topology, and related areas of mathematics, a Stone space is a non-empty compact totally disconnected Hausdorff space. Such spaces are also called profinite spaces. They are named after Marshall Harvey Stone.

In mathematics, a cofinite subset of a set X is a subset A whose complement in X is a finite set. In other words, A contains all but finitely many elements of X. If the complement is not finite, but it is countable, then one says the set is cocountable.

In functional analysis, an operator algebra is an algebra of continuous linear operators on a topological vector space with the multiplication given by the composition of mappings.

In mathematics, a closure operator on a set S is a function from the power set of S to itself which satisfies the following conditions for all sets

In topology, an Alexandrov topology is a topology in which the intersection of any family of open sets is open. It is an axiom of topology that the intersection of any finite family of open sets is open; in Alexandrov topologies the finite restriction is dropped.

In mathematics, Stone's representation theorem for Boolean algebras states that every Boolean algebra is isomorphic to a certain field of sets. The theorem is fundamental to the deeper understanding of Boolean algebra that emerged in the first half of the 20th century. The theorem was first proved by Marshall H. Stone (1936). Stone was led to it by his study of the spectral theory of operators on a Hilbert space.

In abstract algebra, a monadic Boolean algebra is an algebraic structure A with signature

In mathematics a field of sets is a pair where is a set and is an algebra over i.e., a non-empty subset of the power set of closed under the intersection and union of pairs of sets and under complements of individual sets. In other words, forms a subalgebra of the power set Boolean algebra of . Elements of are called points and those of are called complexes and are said to be the admissible sets of .

In mathematics, a complete Boolean algebra is a Boolean algebra in which every subset has a supremum. Complete Boolean algebras are used to construct Boolean-valued models of set theory in the theory of forcing. Every Boolean algebra A has an essentially unique completion, which is a complete Boolean algebra containing A such that every element is the supremum of some subset of A. As a partially ordered set, this completion of A is the Dedekind–MacNeille completion.

The word 'algebra' is used for various branches and structures of mathematics. For their overview, see Algebra.

In mathematical logic, algebraic semantics is a formal semantics based on algebras studied as part of algebraic logic. For example, the modal logic S4 is characterized by the class of topological boolean algebras—that is, boolean algebras with an interior operator. Other modal logics are characterized by various other algebras with operators. The class of boolean algebras characterizes classical propositional logic, and the class of Heyting algebras propositional intuitionistic logic. MV-algebras are the algebraic semantics of Łukasiewicz logic.

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

In mathematics, there are many types of algebraic structures which are studied. Abstract algebra is primarily the study of specific algebraic structures and their properties. Algebraic structures may be viewed in different ways, however the common starting point of algebra texts is that an algebraic object incorporates one or more sets with one or more binary operations or unary operations satisfying a collection of axioms.

In mathematics, the Stone functor is a functor S: TopopBool, where Top is the category of topological spaces and Bool is the category of Boolean algebras and Boolean homomorphisms. It assigns to each topological space X the Boolean algebra S(X) of its clopen subsets, and to each morphism fop: XY in Topop the homomorphism S(f): S(X) → S(Y) given by S(f)(Z) = f−1[Z].