Meaning of sober space | Babel Free

Definitions

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

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.”

“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 […] .”

“Thus sober spaces are necessarily T₀.”

“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.”

“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.”

CEFR level