Theorems · Theorem · general topology
mem_closure_iff
∀ {X : Type u} [inst : TopologicalSpace X] {x : X} {s : Set X},
x ∈ closure s ↔ ∀ (o : Set X), IsOpen o → x ∈ o → (o ∩ s).Nonempty- Defined in
- Mathlib.Topology.Closure
- Cited by
- 15 results in Mathlib
- Foundations
- Depth 62 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.
Cites12
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
- Set.ofPredproof · cited by 6,101
- Compl.complproof · cited by 2,925
- Set.Nonemptystatement and proof · cited by 2,627
- IsOpenstatement and proof · cited by 2,400
- IsClosedproof · cited by 1,639
- closurestatement and proof · cited by 1,254
- IsClosed.isOpen_complproof · cited by 126
- closure_minimalproof · cited by 94
- by_contradictionproof · cited by 42
- isClosed_compl_iffproof · cited by 35
Cited by15
Results whose statement or proof uses this declaration.
- dense_iff_inter_openproof · cited by 13
- specializes_TFAEproof · cited by 6
- continuousWithinAt_right_of_monotoneOn_of_closure_image_mem_nhdsWithinproof · cited by 4
- IsPreconnected.subset_closureproof · cited by 3
- IsOpen.add_closureproof · cited by 3
- IsOpen.mul_closureproof · cited by 2
- closure_inter_open_nonempty_iffproof · cited by 2
- derivedSet_closureproof · cited by 1
- TopologicalSpace.vietoris.closure_finite_subsetsproof · cited by 1
- IsPreirreducible.preimage_of_dense_isPreirreducible_fiberproof · cited by 1
- TopologicalSpace.IsSeparable.univ_piproof · cited by 1