Theorems · Theorem · general topology
mem_connectedComponent
∀ {α : Type u} [inst : TopologicalSpace α] {x : α}, x ∈ connectedComponent x- Defined in
- Mathlib.Topology.Connected.Basic
- Cited by
- 21 results in Mathlib
- Foundations
- Depth 12 from the axioms · uses no axioms
- Assumes
- TopologicalSpace
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites6
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement · cited by 53,352
- TopologicalSpacestatement and proof · cited by 24,529
- Set.mem_singletonproof · cited by 183
- connectedComponentstatement · cited by 68
- Set.mem_sUnion_of_memproof · cited by 11
- isPreconnected_singletonproof · cited by 3
Cited by21
Results whose statement or proof uses this declaration.
- isConnected_connectedComponentproof · cited by 7
- connectedComponent_eqproof · cited by 6
- Continuous.image_connectedComponent_subsetproof · cited by 6
- IsClopen.connectedComponent_subsetproof · cited by 5
- totallyDisconnectedSpace_iff_connectedComponent_singletonproof · cited by 4
- isClosed_connectedComponentproof · cited by 3
- connectedComponentIn_univproof · cited by 3
- connectedComponent_eq_iff_memproof · cited by 2
- Topology.IsCoinducing.preimage_connectedComponentproof · cited by 2
- IsLocallyConstant.of_constant_on_connected_clopensproof · cited by 1
- IsLocallyConstant.of_constant_on_connected_componentsproof · cited by 1
- IsClopen.biUnion_connectedComponent_eqproof · cited by 1