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Concept
Sober space
Summary
In mathematics, a sober space is a topological space X such that every (nonempty) irreducible closed subset of X is the closure of exactly one point of X: that is, every irreducible closed subset has a unique generic point.
Sober spaces have a variety of cryptomorphic definitions, which are documented in this section. All except the definition in terms of nets are described in. In each case below, replacing "unique" with "at most one" gives an equivalent formulation of the T0 axiom. Replacing it with "at least one" is equivalent to the property that the T0 quotient of the space is sober, which is sometimes referred to as having "enough points" in the literature.
A topological space X is sober if every map that preserves all joins and all finite meets from its partially ordered set of open subsets to is the inverse image of a unique continuous function from the one-point space to X.
This may be viewed as a correspondence between the notion of a point in a locale and a point in a topological space, which is the motivating definition.
A filter F of open sets is said to be completely prime if for any family of open sets such that , we have that for some i. A space X is sober if it each completely prime filter is the neighbourhood filter of a unique point in X.
A net is self-convergent if it converges to every point in , or equivalently if its eventuality filter is completely prime. A net that converges to converges strongly if it can only converge to points in the closure of . A space is sober if every self-convergent net converges strongly to a unique point .
In particular, a space is T1 and sober precisely if every self-convergent net is constant.
A closed set is irreducible if it cannot be written as the union of two proper closed subsets. A space is sober if every irreducible closed subset is the closure of a unique point.
A space X is sober if every functor from the category of sheaves Sh(X) to Set that preserves all finite limits and all small colimits must be the stalk functor of a unique point x.
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