Theorems · Theorem · general topology
Continuous.range_subset_closure_image_dense
∀ {X : Type u_1} {Y : Type u_2} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y] {s : Set X} {f : X → Y},
Continuous f → Dense s → Set.range f ⊆ closure (f '' s)- Defined in
- Mathlib.Topology.Continuous
- Cited by
- 7 results in Mathlib
- Foundations
- Depth 77 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites10
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 and proof · cited by 5,609
- Set.rangestatement · cited by 4,705
- Continuousstatement and proof · cited by 2,592
- closurestatement and proof · cited by 1,254
- Densestatement and proof · cited by 359
- Set.image_univproof · cited by 322
- Dense.closure_eqproof · cited by 24
- image_closure_subset_closure_imageproof · cited by 21
Cited by7
Results whose statement or proof uses this declaration.
- DenseRange.dense_imageproof · cited by 10
- Dense.upperBounds_imageproof · cited by 2
- IsDenseInducing.isUniformInducing_extendproof · cited by 1
- cfcₙAux_mem_range_inrproof · cited by 1
- Topology.IsOpenEmbedding.baireSpaceproof · cited by 1
- Dense.ciSup'proof · cited by 1
- Metric.separableSpaceInductiveLimit_of_separableSpaceproof · cited by 0