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Homesober space
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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.

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

Any sober space is T0.

Any Hausdorff space is sober.

A closed subspace of a sober space is sober.

Any product of sober spaces is sober.
Defines:
irreducible set
Type of Math Object:
Definition
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Reference
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