Theorems · Theorem · general topology
IsClosed.isNowhereDense_iff
∀ {X : Type u_1} [inst : TopologicalSpace X] {s : Set X}, IsClosed s → (IsNowhereDense s ↔ interior s = ∅)A closed set is nowhere dense iff its interior is empty.
- Defined in
- Mathlib.Topology.GDelta.Basic
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 63 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- TopologicalSpace
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites6
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement and proof · cited by 53,352
- TopologicalSpacestatement and proof · cited by 24,529
- IsClosedstatement and proof · cited by 1,639
- interiorstatement and proof · cited by 714
- IsClosed.closure_eqproof · cited by 139
- IsNowhereDensestatement · cited by 15
Cited by2
Results whose statement or proof uses this declaration.
- IsNowhereDense.of_isClosed_nullproof · cited by 1
- isClosed_isNowhereDense_iff_complproof · cited by 0