Theorems · Theorem · general topology
IsOpen.connectedComponentIn
∀ {α : Type u} [inst : TopologicalSpace α] [LocallyConnectedSpace α] {F : Set α} {x : α},
IsOpen F → IsOpen (connectedComponentIn F x)- Cited by
- 3 results in Mathlib
- Foundations
- Depth 81 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites11
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
- nhdsproof · cited by 5,554
- IsOpenstatement and proof · cited by 2,400
- IsOpen.mem_nhdsproof · cited by 470
- isOpen_iff_mem_nhdsproof · cited by 48
- connectedComponentInstatement and proof · cited by 33
- LocallyConnectedSpacestatement and proof · cited by 26
- connectedComponentIn_subsetproof · cited by 7
- connectedComponentIn_eqproof · cited by 3
- connectedComponentIn_mem_nhdsproof · cited by 2
Cited by3
Results whose statement or proof uses this declaration.
- isOpen_connectedComponentproof · cited by 3
- locallyConnectedSpace_iff_connectedComponentIn_openproof · cited by 2
- Topology.IsCoinducing.locallyConnectedSpaceproof · cited by 1