Theorems · Theorem · general topology
IsOpen.inter_closure
∀ {X : Type u} [inst : TopologicalSpace X] {s t : Set X}, IsOpen s → s ∩ closure t ⊆ closure (s ∩ t)- Defined in
- Mathlib.Topology.Neighborhoods
- Cited by
- 8 results in Mathlib
- Foundations
- Depth 65 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.
Cites10
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
- Compl.complproof · cited by 2,925
- IsOpenstatement and proof · cited by 2,400
- closurestatement and proof · cited by 1,254
- interiorproof · cited by 714
- Set.compl_subset_complproof · cited by 50
- IsOpen.isClosed_complproof · cited by 50
- Set.compl_interproof · cited by 26
- IsClosed.interior_union_leftproof · cited by 2
Cited by8
Results whose statement or proof uses this declaration.
- Dense.inter_of_isOpen_leftproof · cited by 7
- tangentConeAt_closureproof · cited by 4
- IsOpen.closure_interproof · cited by 2
- Dense.open_subset_closure_interproof · cited by 2
- isLocallyClosed_tfaeproof · cited by 2
- hasFDerivWithinAt_closure_of_tendsto_fderivproof · cited by 2
- TopologicalSpace.Compacts.isPreconnected_nonempty_subsetsproof · cited by 2
- closure_sdiffproof · cited by 1