Zariski space

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In algebraic geometry, a Zariski space, named for Oscar Zariski, has several different meanings:

  • A topological space that is Noetherian (every open set is quasicompact)
  • A topological space that is Noetherian and also sober (every nonempty closed irreducible subset is the closure of a unique point). The spectrum of any commutative Noetherian ring is a Zariski space in this sense
  • A Zariski–Riemann space of valuations of a field