Theorems · Theorem · general topology
closure_eq_interior_union_frontier
∀ {X : Type u} [inst : TopologicalSpace X] (s : Set X), closure s = interior s ∪ frontier s- Defined in
- Mathlib.Topology.Closure
- Cited by
- 4 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.
Cites7
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
- closurestatement · cited by 1,254
- interiorstatement · cited by 714
- frontierstatement · cited by 214
- Set.union_sdiff_cancelproof · cited by 16
- interior_subset_closureproof · cited by 7
Cited by4
Results whose statement or proof uses this declaration.
- Subalgebra.spectrum_sUnion_connectedComponentInproof · cited by 2
- intrinsicInterior_union_intrinsicFrontierproof · cited by 1
- Complex.exists_mem_frontier_isMaxOn_normproof · cited by 1
- IsCompact.exists_mem_frontier_infDist_compl_eq_distproof · cited by 0