# algebraic geometry – Is the Zariski topology on a variety \$V\$ a maximal Noetherian topology?

Let $$K$$ be an algebraically closed field. By a variety $$V$$ definable over $$K$$, I mean a quasi-projective or an algebraic variety in sense of Weil. It is the set of points in an affine or a projective space over some bigger algebraically closed field $$L$$.

Now consider the Zariski topology $$tau$$ on $$V$$ together with Krull dimension on closed sets. Is it possible to enrich this topology in the naive sense by adding new closed sets, so that the enriched topology $$tau^prime$$ is also a Noetherian topology?

In other words, is the Zariski topology on $$V$$ a maximal noetherian topology?