In mathematics and particularly in topology, pairwise Stone space is a bitopological space 
which is pairwise compact, pairwise Hausdorff, and pairwise zero-dimensional.
Mathematics includes the study of such topics as quantity, structure, space, and change.
In mathematics, topology is concerned with the properties of space that are preserved under continuous deformations, such as stretching, twisting, crumpling and bending, but not tearing or gluing.
In mathematics, a bitopological space is a set endowed with two topologies. Typically, if the set is and the topologies are and then the bitopological space is referred to as . The notion was introduced by Kelly in the study of quasimetrics, i.e. distance functions that are notrequired to be symmetric.
Pairwise Stone spaces are a bitopological version of the Stone spaces.
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.
Pairwise Stone spaces are closely related to spectral spaces.
In mathematics, a spectral space is a topological space that is homeomorphic to the spectrum of a commutative ring.
Theorem: [1] If 
is a spectral space, then 
is a pairwise Stone space, where 
is the de Groot dual topology of 
. Conversely, if is a pairwise Stone space, then both 
and 
are spectral spaces.
In mathematics, in particular in topology, the de Groot dual of a topology τ on a set X is the topology τ* whose closed sets are generated by compact saturated subsets of.
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In mathematics, pointless topology is an approach to topology that avoids mentioning points.
In mathematics, and more specifically in topology, an open set is an abstract concept generalizing the idea of an open interval in the real line. The simplest example is in metric spaces, where open sets can be defined as those sets which contain a ball around each of their points ; however, an open set, in general, can be very abstract: any collection of sets can be called open, as long as the union of an arbitrary number of open sets is open, the intersection of a finite number of open sets is open, and the space itself is open. These conditions are very loose, and they allow enormous flexibility in the choice of open sets. In the two extremes, every set can be open, or no set can be open but the space itself and the empty set.
In mathematics, a distributive lattice is a lattice in which the operations of join and meet distribute over each other. The prototypical examples of such structures are collections of sets for which the lattice operations can be given by set union and intersection. Indeed, these lattices of sets describe the scenery completely: every distributive lattice is—up to isomorphism—given as such a lattice of sets.
In mathematics, there is an ample supply of categorical dualities between certain categories of topological spaces and categories of partially ordered sets. Today, these dualities are usually collected under the label Stone duality, since they form a natural generalization of Stone's representation theorem for Boolean algebras. These concepts are named in honor of Marshall Stone. Stone-type dualities also provide the foundation for pointless topology and are exploited in theoretical computer science for the study of formal semantics.
In topology and related areas of mathematics, the set of all possible topologies on a given set forms a partially ordered set. This order relation can be used for comparison of the topologies.
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 topology and related areas of mathematics, a subspace of a topological space X is a subset S of X which is equipped with a topology induced from that of X called the subspace topology.
In mathematics, additivity and sigma additivity of a function defined on subsets of a given set are abstractions of how intuitive properties of size of a set sum when considering multiple objects. Additivity is a weaker condition than σ-additivity, and σ-additivity implies additivity.
In general topology and related areas of mathematics, the initial topology on a set
, with respect to a family of functions on
, is the coarsest topology on X that makes those functions continuous.
In functional analysis and related areas of mathematics a dual topology is a locally convex topology on a dual pair, two vector spaces with a bilinear form defined on them, so that one vector space becomes the continuous dual of the other space.
In functional analysis and related areas of mathematics, the Mackey topology, named after George Mackey, is the finest topology for a topological vector space which still preserves the continuous dual. In other words the Mackey topology does not make linear functions continuous which were discontinuous in the default topology.
In general topology and related areas of mathematics, the final topology on a set
, with respect to a family of functions on
, is the finest topology on
that makes those functions continuous.
In topology and related areas of mathematics, an induced topology on a topological space is a topology which makes the inducing function continuous from/to this topological space.
In mathematics, Birkhoff's representation theorem for distributive lattices states that the elements of any finite distributive lattice can be represented as finite sets, in such a way that the lattice operations correspond to unions and intersections of sets. The theorem can be interpreted as providing a one-to-one correspondence between distributive lattices and partial orders, between quasi-ordinal knowledge spaces and preorders, or between finite topological spaces and preorders. It is named after Garrett Birkhoff, who published a proof of it in 1937.
In mathematics, a Priestley space is an ordered topological space with special properties. Priestley spaces are named after Hilary Priestley who introduced and investigated them. Priestley spaces play a fundamental role in the study of distributive lattices. In particular, there is a duality between the category of Priestley spaces and the category of bounded distributive lattices.
In mathematics, Esakia duality is the dual equivalence between the category of Heyting algebras and the category of Esakia spaces. Esakia duality provides an order-topological representation of Heyting algebras via Esakia spaces.
In mathematics, and especially topology, a Poincaré complex is an abstraction of the singular chain complex of a closed, orientable manifold.
