Theorems · Theorem · general topology

PrimitiveSpectrum.isClosed_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}, IsClosed S ↔ ∃ a, S = PrimitiveSpectrum.hull T a

When α is complete, a set is closed in the relative lower topology if and only if it is of the form hull T a for some a in α.

Defined in

Mathlib.Topology.Order.HullKernel

Cited by

2 results in Mathlib

Foundations

Depth 83 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CompleteLatticeTopologicalSpaceTopology.IsLower