sober space
Let $X$ be a topological space^{}. A subset $A$ of $X$ is said to be irreducible if whenever $A\subseteq B\cup C$ with $B,C$ closed, we have $A\subseteq B$ or $A\subseteq C$. Any singleton and its closure^{} are irreducible. More generally, the closure of an irreducible set is irreducible.
A topological space $X$ is called a sober space if every irreducible closed subset is the closure of some unique point in $X$.
Remarks.

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For any sober space, the closure of a point determines the point. In other words, $\mathrm{cl}(x)=\mathrm{cl}(y)$ implies $x=y$.

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A space is sober iff the closure of every irreducible set is the closure of a unique point.

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Any sober space is T0.

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Any Hausdorff space is sober.

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A closed subspace of a sober space is sober.

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Any product of sober spaces is sober.
Title  sober space 

Canonical name  SoberSpace 
Date of creation  20130322 16:43:44 
Last modified on  20130322 16:43:44 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  6 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 54E99 
Defines  irreducible set 