Stone functor
Encyclopedia
In mathematics
, the Stone functor S: Topop → Bool, where Top is the category of topological spaces
and Bool is the category
of Boolean algebras and Boolean homomorphisms, is the functor
that assigns to each topological space
X the Boolean algebra S(X) of its clopen
subsets, and to each morphism f: X → Y in Topop (i.e., a continuous map f: Y → X) the homomorphism S(f): S(X) → S(Y) given by S(f)(Z) = f−1[Z].
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...
, the Stone functor S: Topop → Bool, where Top is the category of topological spaces
Category of topological spaces
In mathematics, the category of topological spaces, often denoted Top, is the category whose objects are topological spaces and whose morphisms are continuous maps. This is a category because the composition of two continuous maps is again continuous...
and Bool is the category
Category (mathematics)
In mathematics, a category is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose...
of Boolean algebras and Boolean homomorphisms, is the functor
Functor
In category theory, a branch of mathematics, a functor is a special type of mapping between categories. Functors can be thought of as homomorphisms between categories, or morphisms when in the category of small categories....
that assigns to each 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...
X the Boolean algebra S(X) of its clopen
Clopen set
In topology, a clopen set in a topological space is a set which is both open and closed. That this is possible for a set is not as counter-intuitive as it might seem if the terms open and closed were thought of as antonyms; in fact they are not...
subsets, and to each morphism f: X → Y in Topop (i.e., a continuous map f: Y → X) the homomorphism S(f): S(X) → S(Y) given by S(f)(Z) = f−1[Z].