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?
