Theorems · Theorem · general topology
Sigma.isConnected_iff
∀ {ι : Type u_1} {X : ι → Type u_2} [inst : (i : ι) → TopologicalSpace (X i)] {s : Set ((i : ι) × X i)},
IsConnected s ↔ ∃ i t, IsConnected t ∧ s = Sigma.mk i '' t- Defined in
- Mathlib.Topology.Connected.Clopen
- Cited by
- 2 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.
Cites17
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.rangeproof · cited by 4,705
- Continuous.continuousOnproof · cited by 311
- IsConnectedstatement and proof · cited by 116
- Set.image_preimage_eq_of_subsetproof · cited by 41
- IsConnected.isPreconnectedproof · cited by 36
- sigma_mk_injectiveproof · cited by 27
- IsConnected.nonemptyproof · cited by 18
- IsConnected.imageproof · cited by 15
Cited by2
Results whose statement or proof uses this declaration.
- Continuous.exists_lift_sigmaproof · cited by 1
- Sigma.isPreconnected_iffproof · cited by 0