Theorems · Theorem · general topology
IsPreconnected.subset_connectedComponent
∀ {α : Type u} [inst : TopologicalSpace α] {x : α} {s : Set α}, IsPreconnected s → x ∈ s → s ⊆ connectedComponent x- Defined in
- Mathlib.Topology.Connected.Basic
- Cited by
- 10 results in Mathlib
- Foundations
- Depth 8 from the axioms · uses no axioms
- 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
- IsPreconnectedstatement and proof · cited by 205
- connectedComponentstatement · cited by 68
- Set.mem_sUnion_of_memproof · cited by 11
Cited by10
Results whose statement or proof uses this declaration.
- IsConnected.subset_connectedComponentproof · cited by 6
- IsPreconnected.subset_connectedComponentInproof · cited by 6
- connectedComponentIn_univproof · cited by 3
- connectedComponent_eq_iInter_isClopenproof · cited by 2
- totallyDisconnectedSpace_iff_connectedComponent_subsingletonproof · cited by 1
- preconnectedSpace_iff_connectedComponentproof · cited by 1
- connectedComponent_prodproof · cited by 0
- connectedSpace_iff_connectedComponentproof · cited by 0
- IsClopen.connectedComponentIn_eqproof · cited by 0
- connectedComponent_piproof · cited by 0