# Noetherian and Hausdorff space

A Noetherian space satisfies that any non-empty set of open subsets must have a maximal element. But a Hausdorff space satisfies that, for $$x\neq y$$ two different points, there exists open neighborhoods $$U,V$$ containing $$x,y$$ respectively such that $$U\cap V=\emptyset$$.

My question is: when we consider the set of open subsets $$\{U,V\}$$ there is no maximal element in this set. So how can a Noetherian space be Hausdorff too? Thank you!

• Easiest example are finite sets with the discrete topology. Those are both noetherian and Hausdorff. Aug 25, 2022 at 20:16
• And finite discrete spaces are in fact the only examples of Noetherian Hausdorff spaces. Aug 25, 2022 at 21:06

Actually both $$U$$ and $$V$$ are maximal elements of $$\{U,V\}$$. It seems you’re confusing maximal elements with greatest elements. If we have a partially ordered set $$(X,\leq)$$, then an element $$x$$ is called a greatest element if we have $$y\leq x$$ for all $$y$$. It is called a maximal element if for all $$y$$, we have that $$x\leq y$$ implies $$y=x$$. The characterization of Noetherian spaces that you mention talks about maximal elements, not greatest elements.