Theorems · Theorem · general topology
TopologicalSpace.NonemptyCompacts.isClosed_subsets_of_isClosed
∀ {α : Type u_1} [inst : TopologicalSpace α] {F : Set α}, IsClosed F → IsClosed {K | ↑K ⊆ F}- Defined in
- Mathlib.Topology.Sets.VietorisTopology
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 78 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.
Cites9
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
- SetLike.coestatement · cited by 8,199
- Set.ofPredstatement · cited by 6,101
- IsClosedstatement and proof · cited by 1,639
- IsClosed.preimageproof · cited by 138
- TopologicalSpace.NonemptyCompactsstatement · cited by 137
- IsClosed.powerset_vietorisproof · cited by 7
- TopologicalSpace.NonemptyCompacts.continuous_coeproof · cited by 5
Cited by1
Results whose statement or proof uses this declaration.
- TopologicalSpace.NonemptyCompacts.preconnectedSpace_iffproof · cited by 0