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.



Sober topological subspace
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](http://en.wikipedia.org/wiki/Sober_space). 


