Theorems · Theorem · general topology
continuous_generateFrom_iff
∀ {α : Type u} {β : Type v} {f : α → β} {t : TopologicalSpace α} {b : Set (Set β)},
Continuous f ↔ ∀ s ∈ b, IsOpen (f ⁻¹' s)- Defined in
- Mathlib.Topology.Order
- Cited by
- 7 results in Mathlib
- Foundations
- Depth 67 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.ofPredproof · cited by 6,101
- Set.preimagestatement and proof · cited by 4,946
- Continuousstatement · cited by 2,592
- IsOpenstatement and proof · cited by 2,400
- TopologicalSpace.generateFromstatement · cited by 62
- continuous_iff_coinduced_leproof · cited by 13
- TopologicalSpace.le_generateFrom_iff_subset_isOpenproof · cited by 5
Cited by7
Results whose statement or proof uses this declaration.
- OrderIso.continuousproof · cited by 6
- TopologicalSpace.IsTopologicalBasis.continuous_iffproof · cited by 4
- ContinuousMap.continuous_compactOpenproof · cited by 2
- TopologicalSpace.Compacts.continuous_prodproof · cited by 2
- Topology.WithUpper.continuous_toUpperproof · cited by 0
- Topology.WithLower.continuous_toLowerproof · cited by 0
- TopologicalSpace.vietoris.continuous_iffproof · cited by 0