A topological space is sober when every nonempty irreducible closed subset is the closure of a unique point. Every Hausdorff space is sober. In contrast, an infinite set with the cofinite topology has the whole space irreducible and closed, while every singleton is closed, so it is not sober.
The sobrification has one point for every nonempty irreducible closed subset of , with opens . Irreducibility gives , and the map from original opens to these opens is bijective, detected by singleton closures. An irreducible closed subset of the new space has complement ; is an irreducible closed subset of and uniquely generates it. Thus the new space is a sober space.
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"Sober Space" often refers to environments, communities, or forums that promote sobriety and support individuals in recovery from substance use disorders. These spaces are designed to offer a safe, healthy, and stigma-free atmosphere where individuals can connect, share experiences, and receive support in their journey towards sobriety. In practice, sober spaces can include sober living houses, sober bars, support groups like Alcoholics Anonymous (AA), or social events that are alcohol-free.