Sober space

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In mathematics, particularly in topology, a sober space is a particular kind of topological space.

Specifically, a space X is sober if every irreducible closed subset of X is the closure of exactly one singleton of X. An irreducible closed subset of X is a nonempty closed subset of X which is not the union of two proper closed subsets of itself.

Any Hausdorff (T₂) space is sober, and all sober spaces are Kolmogorov (T₀). Sobriety is not comparable to the T₁ condition.

Sobriety of X is precisely a condition that forces the lattice of open subsets of X to determine X up to homeomorphism, which is relevant to pointless topology.

Sobriety makes the specialization preorder a directed complete partial order.

[edit] References

• Discussion of weak separation axioms (PDF file)