Theorems · Theorem · general topology

exists_open_dense_of_open_dense_subtype

∀ {X : Type u} [inst : TopologicalSpace X] {s : Set X},
  Dense s → ∀ {u : Set ↑s}, IsOpen u → Dense u → ∃ v, IsOpen v ∧ Dense v ∧ Subtype.val ⁻¹' v = u

If s is dense in X and u is open and dense in s, then u = v ∩ s for some v that is open and dense in X.

Defined in

Mathlib.Topology.Constructions

Cited by

1 results in Mathlib

Foundations

Depth 72 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

TopologicalSpace