Theorems · Theorem · general topology
continuous_def
∀ {X : Type u_1} {Y : Type u_2} {x : TopologicalSpace X} {x_1 : TopologicalSpace Y} {f : X → Y},
Continuous f ↔ ∀ (s : Set Y), IsOpen s → IsOpen (f ⁻¹' s)- Defined in
- Mathlib.Topology.Continuous
- Cited by
- 21 results in Mathlib
- Foundations
- Depth 6 from the axioms · uses no axioms
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites6
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.preimagestatement and proof · cited by 4,946
- Continuousstatement and proof · cited by 2,592
- IsOpenstatement and proof · cited by 2,400
- Continuous.isOpen_preimageproof · cited by 51
Cited by21
Results whose statement or proof uses this declaration.
- Continuous.compproof · cited by 371
- continuous_idproof · cited by 192
- continuous_iff_continuousAtproof · cited by 139
- continuous_iff_isClosedproof · cited by 24
- continuous_of_discreteTopologyproof · cited by 20
- continuous_iff_coinduced_leproof · cited by 13
- continuousOn_iff'proof · cited by 9
- continuous_id_iff_leproof · cited by 4
- Topology.WithGeneratedByTopology.continuous_equivproof · cited by 4
- Continuous.coinduced_leproof · cited by 4
- Topology.IsGeneratedBy.equiv_symm_comp_continuous_iffproof · cited by 4
- Topology.IsCoherentWith.continuous_iffproof · cited by 3