Sober spaces have a variety of
cryptomorphic definitions, which are documented in this section. 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.
With irreducible closed sets A closed set is
irreducible if it cannot be written as the union of two proper closed subsets. A space is
sober if every nonempty irreducible closed subset is the closure of a unique point. === In terms of morphisms of
frames and locales === A topological space
X is sober if every map from its
partially ordered set of open subsets to that preserves all
joins and all finite meets 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.
Using completely prime filters A
filter F of open sets is said to be
completely prime if for any family O_i of open sets such that \bigcup_i O_i \in F, we have that O_i \in F for some
i. A space
X is sober if each completely prime filter is the
neighbourhood filter of a unique point in
X.
In terms of nets A
net x_{\bullet} is
self-convergent if it converges to every point x_i in x_{\bullet}, or equivalently if its eventuality filter is completely prime. A net x_{\bullet} that converges to x
converges strongly if it can only converge to points in the closure of x. A space is sober if every self-convergent net x_{\bullet} converges strongly to a unique point x. In particular, a space is T1 and sober precisely if every self-convergent net is constant.
As a property of sheaves on the space 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. == Properties and examples ==