Theorems · Theorem · general topology
IsPreconnected.subset_closure
∀ {α : Type u} [inst : TopologicalSpace α] {s t : Set α}, IsPreconnected s → s ⊆ t → t ⊆ closure s → IsPreconnected tTheorem of bark and tree: if a set is within a preconnected set and its closure, then it is
preconnected as well. See also IsConnected.subset_closure.
- Defined in
- Mathlib.Topology.Connected.Basic
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 63 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.
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.Nonemptyproof · cited by 2,627
- IsOpenproof · cited by 2,400
- closurestatement and proof · cited by 1,254
- Set.Subset.transproof · cited by 218
- IsPreconnectedstatement and proof · cited by 205
- mem_closure_iffproof · cited by 15
Cited by3
Results whose statement or proof uses this declaration.
- IsConnected.subset_closureproof · cited by 3
- TopologicalSpace.Compacts.isPreconnected_nonempty_subsetsproof · cited by 2
- IsPreconnected.closureproof · cited by 1