Theorems · Theorem · general topology
IsClopen.connectedComponent_subset
∀ {α : Type u} [inst : TopologicalSpace α] {s : Set α} {x : α}, IsClopen s → x ∈ s → connectedComponent x ⊆ s- Defined in
- Mathlib.Topology.Connected.Clopen
- Cited by
- 5 results in Mathlib
- Foundations
- Depth 63 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
- IsClopenstatement and proof · cited by 189
- connectedComponentstatement · cited by 68
- mem_connectedComponentproof · cited by 21
- isPreconnected_connectedComponentproof · cited by 12
- IsPreconnected.subset_isClopenproof · cited by 4
Cited by5
Results whose statement or proof uses this declaration.
- pathComponent_eq_connectedComponentproof · cited by 2
- connectedComponent_subset_iInter_isClopenproof · cited by 1
- IsClopen.biUnion_connectedComponent_eqproof · cited by 1
- Unitary.mem_pathComponentOne_iffproof · cited by 0
- IsClopen.connectedComponentIn_eqproof · cited by 0