Theorems · Theorem · general topology
Continuous.exists_lift_sigma
∀ {α : Type u} {ι : Type u_1} {X : ι → Type u_2} [inst : TopologicalSpace α] [ConnectedSpace α]
[inst_2 : (i : ι) → TopologicalSpace (X i)] {f : α → (i : ι) × X i},
Continuous f → ∃ i g, Continuous g ∧ f = Sigma.mk i ∘ gA continuous map from a connected space to a disjoint union Σ i, X i can be lifted to one of
the components X i. See also ContinuousMap.exists_lift_sigma for a version with bundled
ContinuousMaps.
- Defined in
- Mathlib.Topology.Connected.Clopen
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 80 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites14
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setproof · cited by 53,352
- TopologicalSpacestatement and proof · cited by 24,529
- Set.imageproof · cited by 5,609
- Set.rangeproof · cited by 4,705
- Continuousstatement and proof · cited by 2,592
- IsConnectedproof · cited by 116
- Set.image_subset_rangeproof · cited by 76
- ConnectedSpacestatement and proof · cited by 37
- Topology.IsEmbedding.continuous_iffproof · cited by 21
- Eq.trans_subsetproof · cited by 16
- Topology.IsEmbedding.sigmaMkproof · cited by 5
- isConnected_rangeproof · cited by 3
Cited by1
Results whose statement or proof uses this declaration.
- ContinuousMap.exists_lift_sigmaproof · cited by 0