Coherent space
Encyclopedia
In proof theory
, a coherent space is a concept introduced in the semantic study of linear logic
.
Let a set C be given. Two subsets S,T ⊆ C are said to be orthogonal, written S ⊥ T, if S ∩ T is ∅ or a singleton. For a family of C-sets (i.e., F ⊆ ℘(C)), the dual of F, written F ⊥, is defined as the set of all C-sets S such that for every T ∈ F, S ⊥ T. A coherent space F over C is a family C-sets for which F = (F ⊥) ⊥.
In topology
, a coherent space is another name for spectral space
. A continuous map between coherent spaces is called coherent if it is spectral.
In Proofs and Types coherent spaces are called coherence spaces. A footnote explains that although in the French original they were espaces cohérents, the coherence space translation was used because spectral space
s are sometimes called coherent spaces.
Proof theory
Proof theory is a branch of mathematical logic that represents proofs as formal mathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented as inductively-defined data structures such as plain lists, boxed lists, or trees, which are constructed...
, a coherent space is a concept introduced in the semantic study of linear logic
Linear logic
Linear logic is a substructural logic proposed by Jean-Yves Girard as a refinement of classical and intuitionistic logic, joining the dualities of the former with many of the constructive properties of the latter...
.
Let a set C be given. Two subsets S,T ⊆ C are said to be orthogonal, written S ⊥ T, if S ∩ T is ∅ or a singleton. For a family of C-sets (i.e., F ⊆ ℘(C)), the dual of F, written F ⊥, is defined as the set of all C-sets S such that for every T ∈ F, S ⊥ T. A coherent space F over C is a family C-sets for which F = (F ⊥) ⊥.
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...
, a coherent space is another name for spectral space
Spectral space
In mathematics, a spectral space is a topological space which is homeomorphic to the spectrum of a commutative ring.-Definition:Let X be a topological space and let K\circ be the set of allquasi-compact and open subsets of X...
. A continuous map between coherent spaces is called coherent if it is spectral.
In Proofs and Types coherent spaces are called coherence spaces. A footnote explains that although in the French original they were espaces cohérents, the coherence space translation was used because spectral space
Spectral space
In mathematics, a spectral space is a topological space which is homeomorphic to the spectrum of a commutative ring.-Definition:Let X be a topological space and let K\circ be the set of allquasi-compact and open subsets of X...
s are sometimes called coherent spaces.