Theorems · Theorem · general topology
Specializes.mem_closed
∀ {X : Type u_1} [inst : TopologicalSpace X] {x y : X} {s : Set X}, x ⤳ y → IsClosed s → x ∈ s → y ∈ s- Defined in
- Mathlib.Topology.Inseparable
- Cited by
- 8 results in Mathlib
- Foundations
- Depth 76 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.
Cites5
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
- Specializesstatement and proof · cited by 176
- specializes_iff_forall_closedproof · cited by 2
Cited by8
Results whose statement or proof uses this declaration.
- IsClosed.stableUnderSpecializationproof · cited by 8
- PrimeSpectrum.isClosedMap_comap_of_isIntegralproof · cited by 2
- IsClosed.not_specializesproof · cited by 2
- TopologicalSpace.vietoris.subset_closure_of_specializesproof · cited by 1
- PrimeSpectrum.denseRange_comap_iff_minimalPrimesproof · cited by 0
- TopologicalSpace.vietoris.specializes_iffproof · cited by 0
- AlgebraicGeometry.Scheme.Hom.dense_smoothLocus_of_perfectFieldproof · cited by 0