Theorems · Theorem · general topology
isClosed_of_mem_irreducibleComponents
∀ {X : Type u_1} [inst : TopologicalSpace X], ∀ s ∈ irreducibleComponents X, IsClosed s[Stacks Tag 004W](https://stacks.math.columbia.edu/tag/004W) ((2))
- Defined in
- Mathlib.Topology.Irreducible
- Cited by
- 9 results in Mathlib
- Foundations
- Depth 72 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.
Cites9
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 · cited by 1,639
- closureproof · cited by 1,254
- LE.le.antisymmproof · cited by 507
- subset_closureproof · cited by 309
- irreducibleComponentsstatement and proof · cited by 35
- closure_eq_iff_isClosedproof · cited by 12
- IsIrreducible.closureproof · cited by 7
Cited by9
Results whose statement or proof uses this declaration.
- closure_sUnion_irreducibleComponents_sdiff_singletonproof · cited by 2
- genericPoints.isGenericPoint_ofComponentproof · cited by 2
- PrimeSpectrum.zeroLocus_minimalPrimesproof · cited by 1
- mem_of_subset_sUnion_irreducibleComponentsproof · cited by 1
- irreducibleComponents_eq_maximals_closedproof · cited by 1
- isClosed_irreducibleComponentproof · cited by 1
- AlgebraicGeometry.Scheme.Hom.dense_smoothLocus_of_perfectFieldproof · cited by 0