Theorems · Theorem · general topology
connectedComponentIn_mono
∀ {α : Type u} [inst : TopologicalSpace α] (x : α) {F G : Set α},
F ⊆ G → connectedComponentIn F x ⊆ connectedComponentIn G x- Defined in
- Mathlib.Topology.Connected.Basic
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 80 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- TopologicalSpace
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites14
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.Elemproof · cited by 7,166
- Set.imageproof · cited by 5,609
- Set.image_monoproof · cited by 197
- Set.inclusionproof · cited by 145
- Set.image_compproof · cited by 142
- Set.empty_subsetproof · cited by 70
- connectedComponentproof · cited by 68
- connectedComponentInstatement and proof · cited by 33
- continuous_inclusionproof · cited by 8
- Continuous.image_connectedComponent_subsetproof · cited by 6
Cited by2
Results whose statement or proof uses this declaration.
- ContinuousOn.preimage_connectedComponentInproof · cited by 2
- AmpleSet.unionproof · cited by 0