A generalized Alexandroff space that is not an Alexandroff space

Question

A topological space $(X,T)$ is called generalized Alexandroff if any intersection of open sets is generalized open (where $A$ is generalized open if its interior contains all closed subsets of $A$). It's clear from definition that every Alexandroff space is generalized Alexandroff. But the converse needn't be true, I need an example to show this.

Answer 1

Let $X=\omega_1+1$. The open sets of $X$ are $\varnothing$ and the sets $[\alpha,\omega_1]$ for $\alpha<\omega_1$. Every intersection of open sets is either open or $\{\omega_1\}$, which is generalized open but not open.