Theorems · Theorem · general topology
IsConnected.image
∀ {α : Type u} {β : Type v} [inst : TopologicalSpace α] [inst_1 : TopologicalSpace β] {s : Set α},
IsConnected s → ∀ (f : α → β), ContinuousOn f s → IsConnected (f '' s)The image of a connected set is connected as well.
- Defined in
- Mathlib.Topology.Connected.Basic
- Cited by
- 15 results in Mathlib
- Foundations
- Depth 78 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites9
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 · cited by 5,609
- ContinuousOnstatement and proof · cited by 1,411
- IsConnectedstatement and proof · cited by 116
- IsConnected.isPreconnectedproof · cited by 36
- Set.image_nonemptyproof · cited by 29
- IsPreconnected.imageproof · cited by 24
- IsConnected.nonemptyproof · cited by 18
Cited by15
Results whose statement or proof uses this declaration.
- Orientation.oangle_sign_smul_add_rightproof · cited by 6
- Continuous.image_connectedComponent_subsetproof · cited by 6
- AffineSubspace.isConnected_setOfPred_sSameSideproof · cited by 3
- Real.Angle.sign_eq_of_continuousOnproof · cited by 3
- AffineSubspace.isConnected_setOfPred_sOppSideproof · cited by 2
- AffineSubspace.isConnected_setOfPred_wOppSideproof · cited by 2
- AffineSubspace.isConnected_setOfPred_wSameSideproof · cited by 2
- Collinear.oangle_sign_of_sameRay_vsubproof · cited by 2
- Sigma.isConnected_iffproof · cited by 2
- AffineSubspace.SSameSide.oangle_sign_eqproof · cited by 2
- isConnected_setOfPred_sameRay_and_ne_zeroproof · cited by 2