Theorems · Theorem · general topology
IsPreconnected.subset_connectedComponentIn
∀ {α : Type u} [inst : TopologicalSpace α] {s : Set α} {x : α} {F : Set α},
IsPreconnected s → x ∈ s → s ⊆ F → s ⊆ connectedComponentIn F x- Defined in
- Mathlib.Topology.Connected.Basic
- Cited by
- 6 results in Mathlib
- Foundations
- Depth 79 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.
Cites16
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.preimageproof · cited by 4,946
- le_reflproof · cited by 2,061
- Set.Subset.transproof · cited by 218
- IsPreconnectedstatement and proof · cited by 205
- Set.image_monoproof · cited by 197
- Set.mem_preimageproof · cited by 190
- connectedComponentproof · cited by 68
- Topology.IsInducing.subtypeValproof · cited by 43
- Set.inter_eq_rightproof · cited by 41
- Subtype.image_preimage_coeproof · cited by 37
Cited by6
Results whose statement or proof uses this declaration.
- connectedComponentIn_univproof · cited by 3
- ContinuousOn.image_connectedComponentIn_subsetproof · cited by 3
- connectedComponentIn_mem_nhdsproof · cited by 2
- locallyConnectedSpace_iff_connected_subsetsproof · cited by 2
- IsPreconnected.connectedComponentInproof · cited by 2
- IsClopen.connectedComponentIn_eqproof · cited by 0