sober space

English

Noun

sober space (plural sober spaces)

  1. (topology) A topological space of which every join-irreducible closed subset is the closure of exactly one point of the space.
    • 1975, Mathematica Scandinavica, Societates Mathematicae, page 318,
      For a reader more interested in function spaces than in functors, this concludes the description of content except to add that the classification of coadjoint G’s by sets B bearing a topological topology relativizes to T0 spaces. For other readers: and trivially to sober spaces.
    • 1983, Houston Journal of Mathematics, Volume 9, University of Houston, page 192,
      In the Hofmann and Lawson paper, it is proved that the topological space Spec(L) is a locally quasicompact sober space […] .
    • 2002, P. T. Johnstone, Sketches of an Elephant: A Topos Theory Compendium, Volume 2, Oxford University Press, page 492,
      Thus sober spaces are necessarily T0.
    • 2003, A. Pultr, S. E. Rodabaugh, Chapter 6: Lattice-Valued Frames, Functor Categories, And Classes of Sober Spaces, Stephen Ernest Rodabaugh, Erich Peter Klement (editors), Topological and Algebraic Structures in Fuzzy Sets: A Handbook of Recent Developments in the Mathematics of Fuzzy Sets, Springer, page 155,
      How rich are sober and L-sober spaces, are there important examples? It is well-known [25] that Hausdorff spaces, and hence compact T0 (and so finite T0 spaces), are sober spaces in the traditional setting. Furthermore, the soberification of a space—the spectrum of the topology of a space—is always sober; and if the original space is not Hausdorff, then its soberification is a sober space which is not Hausdorff. So there are many non-Hausdorff sober spaces as well.
    • 2004, Ofer Gabber, Notes on some t-structures, Alan Adolphson, Francesco Baldassarri, Pierre Berthelot, Nicholas Katz, François Loeser (editors) Geometric Aspects of Dwork Theory, Volume 1, Walter de Gruyter, page 711,
      When X is a noetherian sober space the construction of loc. cit. can be extended to an arbitrary lower-semicontinuous function for the constructible topology.

Hypernyms

Hyponyms
  • soberification / sobrification (latter form apparently more common)

Translations

Further reading
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