De Groot dual
Encyclopedia
In mathematics
, in particular in topology
, the de Groot dual (after Johannes de Groot
) of a topology τ on a set X is the topology τ* whose closed sets are generated by compact
saturated
subset
s of (X, τ).
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...
, in particular in topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
, the de Groot dual (after Johannes de Groot
Johannes De Groot
Johannes de Groot was a Dutch mathematician, the leading Dutch topologist for more than two decades following World War II.-Biography:...
) of a topology τ on a set X is the topology τ* whose closed sets are generated by compact
Compact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
saturated
Saturated set
In mathematics, in particular in topology, a subset of a topological space is saturated if it is an intersection of open subsets of X...
subset
Subset
In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
s of (X, τ).