Theorems · Theorem · general topology
TopologicalSpace.Compacts.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
- 7 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.
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
- SetLike.coestatement · cited by 8,199
- Set.ofPredstatement · cited by 6,101
- IsOpenstatement and proof · cited by 2,400
- TopologicalSpace.Compactsstatement · cited by 386
- Set.powersetproof · cited by 67
- Continuous.isOpen_preimageproof · cited by 51
- IsOpen.powerset_vietorisproof · cited by 11
- TopologicalSpace.Compacts.continuous_coeproof · cited by 4
Cited by7
Results whose statement or proof uses this declaration.
- TopologicalSpace.Compacts.isOpen_setOfPred_disjoint_coeproof · cited by 3
- TopologicalSpace.IsTopologicalBasis.compactsproof · cited by 2
- TopologicalSpace.Compacts.continuous_prodproof · cited by 2
- TopologicalSpace.Compacts.isCompact_biUnion_coe_of_isCompactproof · cited by 2
- Topology.IsOpenEmbedding.compacts_mapproof · cited by 1
- TopologicalSpace.Compacts.separableSpace_iffproof · cited by 1
- TopologicalSpace.Compacts.isClosed_inter_nonempty_of_isClosedproof · cited by 0