Theorems · Theorem · general topology
IsConnected.nonempty
∀ {α : Type u} [inst : TopologicalSpace α] {s : Set α}, IsConnected s → s.Nonempty- Defined in
- Mathlib.Topology.Connected.Basic
- Cited by
- 18 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.
Cites4
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.Nonemptystatement · cited by 2,627
- IsConnectedstatement and proof · cited by 116
Cited by18
Results whose statement or proof uses this declaration.
- IsConnected.imageproof · cited by 15
- MeromorphicOn.exists_meromorphicOrderAt_ne_top_iff_forallproof · cited by 7
- Topology.IsCoinducing.isConnected_preimage_of_isClosedproof · cited by 3
- IsConnected.preimage_of_isOpenMapproof · cited by 2
- Topology.IsCoinducing.image_connectedComponentproof · cited by 2
- Subtype.connectedSpaceproof · cited by 2
- exists_eq_const_mul_setIntegral_of_ae_nonnegproof · cited by 2
- Sigma.isConnected_iffproof · cited by 2
- IsConnected.iUnion_of_reflTransGenproof · cited by 1
- eq_Icc_csInf_csSup_of_connected_bdd_closedproof · cited by 1
- Sum.isConnected_iffproof · cited by 1
- AnalyticOnNhd.eq_const_of_im_eq_constproof · cited by 1