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 nonempty irreducible closed subset has a unique generic point.

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.

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.

A topological space X is sober if every map from its partially ordered set of open subsets to ${\displaystyle \{0,1\}}$ 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.

A filter F of open sets is said to be completely prime if for any family ${\displaystyle O_{i}}$ of open sets such that ${\displaystyle \bigcup _{i}O_{i}\in F}$, we have that ${\displaystyle O_{i}\in F}$ for some i. A space X is sober if each completely prime filter is the neighbourhood filter of a unique point in X.

A net ${\displaystyle x_{\bullet }}$ is self-convergent if it converges to every point ${\displaystyle x_{i}}$ in ${\displaystyle x_{\bullet }}$, or equivalently if its eventuality filter is completely prime. A net ${\displaystyle x_{\bullet }}$ that converges to ${\displaystyle x}$ converges strongly if it can only converge to points in the closure of ${\displaystyle x}$. A space is sober if every self-convergent net ${\displaystyle x_{\bullet }}$ converges strongly to a unique point ${\displaystyle x}$.

In particular, a space is T₁ and sober precisely if every self-convergent net is constant.