mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, in particular in

topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...

, the de Groot dual (after

Johannes de Groot Johannes de Groot (May 7, 1914 – September 11, 1972) was a Dutch mathematician, the leading Dutch topologist for more than two decades following World War II.. Biography De Groot was born at Garrelsweer, a village in the municipality of Loppe ...

) of a topology ''τ'' on a

set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...

''X'' is the topology ''τ''* whose

closed sets In geometry, topology, and related branches of mathematics, a closed set is a Set (mathematics), set whose complement (set theory), complement is an open set. In a topological space, a closed set can be defined as a set which contains all its lim ...

are generated by

compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...

saturated Saturation, saturated, unsaturation or unsaturated may refer to: Chemistry * Saturation, a property of organic compounds referring to carbon-carbon bonds ** Saturated and unsaturated compounds **Degree of unsaturation ** Saturated fat or fatty ac ...

subset In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...

s of (''X'', ''τ'').

References

* R. Kopperman (1995), Asymmetry and duality in topology. ''Topology Applications'', 66(1), 1–39, 1995. Topology {{topology-stub