Theorems · Theorem · general topology
image_closure_subset_closure_image
∀ {X : Type u_1} {Y : Type u_2} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y] {f : X → Y} {s : Set X},
Continuous f → f '' closure s ⊆ closure (f '' s)See also IsClosedMap.closure_image_eq_of_continuous.
- Defined in
- Mathlib.Topology.Continuous
- Cited by
- 21 results in Mathlib
- Foundations
- Depth 76 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.imagestatement · cited by 5,609
- Continuousstatement and proof · cited by 2,592
- closurestatement · cited by 1,254
- Set.mapsTo_imageproof · cited by 71
- Set.MapsTo.image_subsetproof · cited by 49
- Set.MapsTo.closureproof · cited by 10
Cited by21
Results whose statement or proof uses this declaration.
- Continuous.range_subset_closure_image_denseproof · cited by 7
- MeasureTheory.SimpleFunc.tendsto_approxOnproof · cited by 6
- IsClosedMap.closure_image_eq_of_continuousproof · cited by 6
- AlgebraicGeometry.Scheme.Hom.support_kerproof · cited by 6
- closure_image_closureproof · cited by 3
- LinearPMap.closure_inverse_graphproof · cited by 2
- Submodule.topologicalClosure_mapproof · cited by 2
- Complex.subfield_eq_of_closedproof · cited by 2
- segment_subset_closure_openSegmentproof · cited by 2
- LinearPMap.inverse_isClosable_iffproof · cited by 1
- Balanced.closureproof · cited by 1
- StarSubalgebra.map_topologicalClosure_leproof · cited by 1