Theorems · Theorem · general topology
TopologicalSpace.NonemptyCompacts.isOpen_subsets_of_isOpen
∀ {α : Type u_1} [inst : TopologicalSpace α] {U : Set α}, IsOpen U → IsOpen {K | ↑K ⊆ U}- Defined in
- Mathlib.Topology.Sets.VietorisTopology
- Cited by
- 4 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
- IsOpenstatement and proof · cited by 2,400
- IsOpen.preimageproof · cited by 147
- TopologicalSpace.NonemptyCompactsstatement · cited by 137
- IsOpen.powerset_vietorisproof · cited by 11
- TopologicalSpace.NonemptyCompacts.continuous_coeproof · cited by 5
Cited by4
Results whose statement or proof uses this declaration.
- TopologicalSpace.NonemptyCompacts.locallyCompactSpace_iffproof · cited by 1
- TopologicalSpace.NonemptyCompacts.locallyConnectedSpace_iffproof · cited by 1
- TopologicalSpace.IsTopologicalBasis.nonemptyCompactsproof · cited by 1
- TopologicalSpace.NonemptyCompacts.preconnectedSpace_iffproof · cited by 0