Theorems · Theorem · general topology

PrimitiveSpectrum.isOpen_iff

∀ {α : Type u_1} [inst : CompleteLattice α] {T : Set α} [inst_1 : TopologicalSpace α] [Topology.IsLower α],
  (∀ p ∈ T, InfPrime p) → ∀ (S : Set ↑T), IsOpen S ↔ ∃ a, S = (PrimitiveSpectrum.hull T a)ᶜ

When α is complete, a set is Lower topology relative-open if and only if it is of the form (hull T a)ᶜ for some a in α.

Defined in

Mathlib.Topology.Order.HullKernel

Cited by

1 results in Mathlib

Foundations

Depth 82 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CompleteLatticeTopologicalSpaceTopology.IsLower