Theorems · Theorem · general topology
continuous_iff_isClosed
∀ {X : Type u_1} {Y : Type u_2} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y] {f : X → Y},
Continuous f ↔ ∀ (s : Set Y), IsClosed s → IsClosed (f ⁻¹' s)- Defined in
- Mathlib.Topology.Continuous
- Cited by
- 24 results in Mathlib
- Foundations
- Depth 57 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites8
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 · cited by 2,592
- IsClosedstatement and proof · cited by 1,639
- Function.Surjective.forallproof · cited by 214
- compl_surjectiveproof · cited by 26
- continuous_defproof · cited by 21
Cited by24
Results whose statement or proof uses this declaration.
- IsClosed.preimageproof · cited by 138
- isClosed_eqproof · cited by 71
- isClosed_leproof · cited by 32
- SeqContinuous.continuousproof · cited by 2
- uCompactlyGeneratedSpace_of_isClosedproof · cited by 2
- MeasureTheory.Measure.haar.is_left_invariant_addCHaarproof · cited by 1
- MeasureTheory.Measure.haar.is_left_invariant_chaarproof · cited by 1
- MeasureTheory.Measure.haar.addCHaar_selfproof · cited by 1
- BoundedContinuousFunction.arzela_ascoli₂proof · cited by 1
- continuousOn_iff_isClosedproof · cited by 1
- preimage_closedPoints_subsetproof · cited by 1
- IsClosed.isGeneratedByproof · cited by 1