World Library  
Flag as Inappropriate
Email this Article

Sober space

Article Id: WHEBN0000602490
Reproduction Date:

Title: Sober space  
Author: World Heritage Encyclopedia
Language: English
Subject: Pointless topology, Specialization (pre)order, Glossary of topology, Spectral space, Stone duality
Publisher: World Heritage Encyclopedia

Sober space

In mathematics, a sober space is a topological space such that every irreducible closed subset of X is the closure of exactly one point of X: that is, this closed subset has a unique generic point.


  • Properties and examples 1
  • See also 2
  • References 3
  • Further reading 4
  • External links 5

Properties and examples

Any Hausdorff (T2) space is sober (the only irreducible subsets being points), and all sober spaces are Kolmogorov (T0), and both implications are strict.[1] Sobriety is not comparable to the T1 condition: an example of a T1 space which is not sober is an infinite set with the cofinite topology, the whole space being an irreducible closed subset with no generic point. Moreover T2 is stronger than T1 and sober, i.e., while every T2 space is at once T1 and sober, there exist spaces that are simultaneously T1 and sober, but not T2. One such example is the following: let X be the set of real numbers, with a new point p adjoined; the open sets being all real open sets, and all cofinite sets containing p.

Sobriety of X is precisely a condition that forces the lattice of open subsets of X to determine X up to homeomorphism, which is relevant to pointless topology.

Sobriety makes the specialization preorder a directed complete partial order.

The prime spectrum Spec(R) of a commutative ring R with the Zariski topology is a compact sober T0 space.[1] In fact, every spectral space (i.e. a compact sober space for which the collection of compact open subsets is closed under finite intersections and forms a base for the topology) is homeomorphic to Spec(R) for some commutative ring R. This is a theorem of Melvin Hochster.[2] More generally, the underlying topological space of any scheme is a sober space.

See also


  1. ^ a b Hart, Klaas Pieter; Nagata, Jun-iti; Vaughan, Jerry E. (2004). Encyclopedia of general topology. Elsevier. pp. 155–156.  
  2. ^ Hochster, Melvin (1969), "Prime ideal structure in commutative rings", Trans. Amer. Math. Soc. 142: 43–60,  

Further reading

  • Pedicchio, Maria Cristina; Tholen, Walter, eds. (2004). Categorical foundations. Special topics in order, topology, algebra, and sheaf theory. Encyclopedia of Mathematics and Its Applications 97. Cambridge:  

External links

  • Discussion of weak separation axioms

This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.

Copyright © World Library Foundation. All rights reserved. eBooks from Project Gutenberg are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.