Theorems · Theorem · general topology
Continuous.closure_preimage_subset
∀ {X : Type u_1} {Y : Type u_2} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y] {f : X → Y},
Continuous f → ∀ (t : Set Y), closure (f ⁻¹' t) ⊆ f ⁻¹' closure t- Defined in
- Mathlib.Topology.Continuous
- Cited by
- 12 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.
Cites11
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
- closurestatement and proof · cited by 1,254
- subset_closureproof · cited by 309
- isClosed_closureproof · cited by 195
- IsClosed.closure_eqproof · cited by 139
- IsClosed.preimageproof · cited by 138
- closure_monoproof · cited by 133
- Set.preimage_monoproof · cited by 95
Cited by12
Results whose statement or proof uses this declaration.
- IsOpenMap.preimage_closure_eq_closure_preimageproof · cited by 12
- Continuous.frontier_preimage_subsetproof · cited by 2
- LinearPMap.closure_inverse_graphproof · cited by 2
- tsupport_comp_subset_preimageproof · cited by 1
- TopologicalSpace.IsOpenCover.quasiSober_iff_forallproof · cited by 1
- Continuous.preimage_coborder_subsetproof · cited by 1
- NumberField.mixedEmbedding.fundamentalCone.closure_normLeOne_subsetproof · cited by 1
- Topology.IsUpperSet.monotone_iff_continuousproof · cited by 1
- intrinsicClosure_subset_closureproof · cited by 0
- closure_ball_subsetproof · cited by 0
- mulTSupport_comp_subset_preimageproof · cited by 0
- intrinsicClosure_monoproof · cited by 0