In terms of morphisms of frames and locales

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 ${\displaystyle \{0,1\}}$ 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 ${\displaystyle O_{i}}$ 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 it each completely prime filter is the neighbourhood filter of a unique point in X.

In terms of nets

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 ${\displaystyle x}$.^[2]

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

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 irreducible closed subset is the closure of a unique point.

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.