sober space

plural sober spaces

1. mathematicsA topological space of which every join-irreducible closed subset is the closure of exactly one point of the space.

   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 […] .

   4 more examples

   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.

   1975, Mathematica Scandinavica, Societates Mathematicae, page 318

   Thus sober spaces are necessarily T₀.

   2002, P. T. Johnstone, Sketches of an Elephant: A Topos Theory Compendium, volume 2, Oxford University Press, page 492

   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.

   2003, A. Pultr, S. E. Rodabaugh, “Chapter 6: Lattice-Valued Frames, Functor Categories, And Classes of Sober Spaces”, in 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

   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.

   2004, Ofer Gabber, “Notes on some t-structures”, in Alan Adolphson, Francesco Baldassarri, Pierre Berthelot, Nicholas Katz, François Loeser, editors, Geometric Aspects of Dwork Theory, volume 1, Walter de Gruyter, page 711