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.

Sketches of an elephant, Part C]. $\endgroup$ – Zhen Lin Jun 22 '14 at 17:08