Theorems · Theorem · general topology
isConnected_univ_pi
∀ {ι : Type u_1} {X : ι → Type u_2} [inst : (i : ι) → TopologicalSpace (X i)] {s : (i : ι) → Set (X i)},
IsConnected (Set.univ.pi s) ↔ ∀ (i : ι), IsConnected (s i)- Defined in
- Mathlib.Topology.Connected.Basic
- Cited by
- 0 results in Mathlib
- Foundations
- Depth 91 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- TopologicalSpace
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Cites12
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.univstatement and proof · cited by 3,945
- Set.Nonemptyproof · cited by 2,627
- Set.pistatement and proof · cited by 405
- Continuous.continuousOnproof · cited by 311
- IsPreconnectedproof · cited by 205
- continuous_applyproof · cited by 120
- IsConnectedstatement · cited by 116
- IsPreconnected.imageproof · cited by 24
- Set.eval_image_univ_piproof · cited by 4
- isPreconnected_univ_piproof · cited by 4
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