Spectral space

In mathematics, a spectral space is a topological space that is homeomorphic to the spectrum of a commutative ring. It is sometimes also called a coherent space because of the connection to coherent topos.

Definition

Let X be a topological space and let K${\displaystyle \circ }$ (X) be the set of all compact open subsets of X. Then X is said to be spectral if it satisfies all of the following conditions:

Equivalent descriptions

Let X be a topological space. Each of the following properties are equivalent to the property of X being spectral:

1. X is homeomorphic to a projective limit of finite T0-spaces.
2. X is homeomorphic to the spectrum of a bounded distributive lattice L. In this case, L is isomorphic (as a bounded lattice) to the lattice K${\displaystyle \circ }$ (X) (this is called Stone representation of distributive lattices).
3. X is homeomorphic to the spectrum of a commutative ring.
4. X is the topological space determined by a Priestley space.
5. X is a T0 space whose frame of open sets is coherent (and every coherent frame comes from a unique spectral space in this way).

Properties

Let X be a spectral space and let K${\displaystyle \circ }$ (X) be as before. Then:

• K${\displaystyle \circ }$ (X) is a bounded sublattice of subsets of X.
• Every closed subspace of X is spectral.
• An arbitrary intersection of compact and open subsets of X (hence of elements from K${\displaystyle \circ }$ (X)) is again spectral.
• X is T0 by definition, but in general not T1.[1] In fact a spectral space is T1 if and only if it is Hausdorff (or T2) if and only if it is a boolean space if and only if K${\displaystyle \circ }$ (X) is a boolean algebra.
• X can be seen as a pairwise Stone space.[2]

Spectral maps

A spectral map f: X → Y between spectral spaces X and Y is a continuous map such that the preimage of every open and compact subset of Y under f is again compact.

The category of spectral spaces, which has spectral maps as morphisms, is dually equivalent to the category of bounded distributive lattices (together with morphisms of such lattices).[3] In this anti-equivalence, a spectral space X corresponds to the lattice K${\displaystyle \circ }$ (X).

Citations

1. ^ A.V. Arkhangel'skii, L.S. Pontryagin (Eds.) General Topology I (1990) Springer-Verlag ISBN 3-540-18178-4 (See example 21, section 2.6.)
2. ^ G. Bezhanishvili, N. Bezhanishvili, D. Gabelaia, A. Kurz, (2010). "Bitopological duality for distributive lattices and Heyting algebras." Mathematical Structures in Computer Science, 20.
3. ^