Theorems · Theorem · general topology
IsPreconnected.image
∀ {α : Type u} {β : Type v} [inst : TopologicalSpace α] [inst_1 : TopologicalSpace β] {s : Set α},
IsPreconnected s → ∀ (f : α → β), ContinuousOn f s → IsPreconnected (f '' s)The image of a preconnected set is preconnected as well.
- Defined in
- Mathlib.Topology.Connected.Basic
- Cited by
- 24 results in Mathlib
- Foundations
- Depth 77 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites19
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.imagestatement and proof · cited by 5,609
- Set.preimageproof · cited by 4,946
- Set.Nonemptyproof · cited by 2,627
- IsOpenproof · cited by 2,400
- ContinuousOnstatement and proof · cited by 1,411
- Set.inter_commproof · cited by 291
- Set.Subset.rflproof · cited by 255
- IsPreconnectedstatement and proof · cited by 205
- Set.image_subset_iffproof · cited by 203
- Set.subset_interproof · cited by 74
Cited by24
Results whose statement or proof uses this declaration.
- IsConnected.imageproof · cited by 15
- Topology.IsInducing.isPreconnected_imageproof · cited by 11
- isPreconnected_iff_preconnectedSpaceproof · cited by 4
- isPreconnected_univ_piproof · cited by 4
- ContinuousOn.image_connectedComponentIn_subsetproof · cited by 3
- IsPreconnected.prodproof · cited by 3
- ContinuousOn.image_Iccproof · cited by 3
- IsPreconnected.mapsTo_Ioi_or_Iioproof · cited by 2
- Real.convexOn_Gammaproof · cited by 2
- TopologicalSpace.Compacts.isPreconnected_Iccproof · cited by 2
- Homeomorph.isPreconnected_imageproof · cited by 2
- isPreconnected_rangeproof · cited by 2