Math 2.0

General topology Definition

Spectral space

Builds on Topological spaceCompact spaceBasis of a topologySober space

Definition

Let XX be a topological space. A subset of XX is compact if every cover of it by open sets of XX has a finite subcover; no Hausdorff condition is assumed. The space XX is spectral if:

  1. XX is compact;
  2. the compact open subsets of XX form a basis of the topology of XX;
  3. the intersection of any two compact open subsets of XX is compact;
  4. XX is sober: every irreducible closed subset of XX is the closure of exactly one point.

Here a closed set is irreducible if it is nonempty and is not the union of two closed proper subsets.

Remarks

A sober space is T0, so every spectral space is T0.

Example: the prime spectrum Spec⁡ℤ\operatorname{Spec} \Z with the Zariski topology. Its open sets are ∅\emptyset and the complements of finite sets of nonzero primes, each of which is compact. The irreducible closed sets are the points (p)(p) for primes pp and the whole space, which is the closure of (0)(0).

Non-example: the interval [0,1][0, 1] with its usual topology. It is compact and sober, but it is connected, so its only compact open subsets are ∅\emptyset and [0,1][0, 1], and these do not form a basis; clause 2 fails.

Hochster's theorem says that the spectral spaces are exactly the spaces homeomorphic to the spectrum of a commutative ring. A Hausdorff spectral space is the same as a Stone space.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

Show the path as a graph
Basis of a topology, concept, no evidence levelBasis of a topologyconceptCompact space, concept, no evidence levelCompact spaceconceptSober space, concept, no evidence levelSober spaceconceptTopological space, concept, no evidence levelTopological spaceconceptSpectral space, definition, no evidence levelSpectral spacedefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Basis of a topology concept
  2. Compact space concept
  3. Sober space concept
  4. Topological space concept
  5. Spectral space definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Spectral_space
About this page
TagQYZXXXFM53
URIm2://QYZXXXFM53
Kinddefinition
WritingW0 stub
MSC54F65, 06D50, 14A05
DefinesSpectral space
UsesTopological space, Compact space, Basis of a topology, Sober space
Changed2 November 2026

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • a source with a locator (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

No page links to this page yet.