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Assume $X$ to be a Notherian topological space such that any irreducible closed subset has a unique generic point. Consider $Y\subseteq X$ as a topological space with the induced topology from $X$. Is it true that any irreducible closed subset of $Y$ has a unique generic point?

What properties on $Y$ make above question to be true?

Any information even in the case that $X= \text{Spec R}$ with the Zariski topology (R is a Notherian ring) is also useful.

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    $\begingroup$ Open subspaces of sober spaces are sober, as are closed subspaces. In general, not every subspace is sober. See Lemma 1.2.5 in [Sketches of an elephant, Part C]. $\endgroup$
    – Zhen Lin
    Commented Jun 22, 2014 at 17:08

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Surely not: the subspace of $\text{Spec}(\mathbb{Z})$ consisting of nonzero primes, which is homeomorphic to $\mathbb{N}$ with the cofinite topology, is not sober. See also Wikipedia.

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  • $\begingroup$ So if $M$ is not a finitely generated module, we still can not say any thing about being Sober of $Y=\text{Supp}_RM$ as a topological subspace of $X=\text{Spec} R$. $\endgroup$
    – user49402
    Commented Jun 22, 2014 at 18:21
  • $\begingroup$ Yes, for example the sum of all $\mathbb{Z}/(p)$ has support given by the nonzero primes $p$. $\endgroup$
    – Todd Trimble
    Commented Jun 22, 2014 at 19:01
  • $\begingroup$ Is this true if $Y$ is an intersection of constructible subsets (i.e. closed sets in the constructible topology)? $\endgroup$
    – Z. M
    Commented Dec 29, 2021 at 20:05

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