Theorems · Theorem · general topology
connectedComponent_subset_iInter_isClopen
∀ {α : Type u} [inst : TopologicalSpace α] {x : α}, connectedComponent x ⊆ ⋂ Z, ↑ZThe connected component of a point is always a subset of the intersection of all its clopen neighbourhoods.
- Defined in
- Mathlib.Topology.Connected.Clopen
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 64 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
- Set.iInterstatement · cited by 1,084
- IsClopenstatement and proof · cited by 189
- connectedComponentstatement · cited by 68
- Set.subset_iInterproof · cited by 17
- IsClopen.connectedComponent_subsetproof · cited by 5
Cited by1
Results whose statement or proof uses this declaration.
- connectedComponent_eq_iInter_isClopenproof · cited by 2