Theorems · Theorem · general topology
IsOpen.mem_nhdsSet
∀ {X : Type u_2} [inst : TopologicalSpace X] {s t : Set X}, IsOpen s → (s ∈ nhdsSet t ↔ t ⊆ s)- Defined in
- Mathlib.Topology.NhdsSet
- Cited by
- 17 results in Mathlib
- Foundations
- Depth 64 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.
Cites7
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
- Filterstatement · cited by 8,121
- IsOpenstatement and proof · cited by 2,400
- nhdsSetstatement · cited by 267
- IsOpen.interior_eqproof · cited by 58
- subset_interior_iff_mem_nhdsSetproof · cited by 11
Cited by17
Results whose statement or proof uses this declaration.
- generalized_tube_lemmaproof · cited by 6
- lebesgue_number_of_compact_openproof · cited by 2
- Metric.thickening_mem_nhdsSetproof · cited by 2
- Ioi_mem_nhdsSet_Ici_iffproof · cited by 2
- IsClosedMap.isEvenlyCovered_of_openPartialHomeomorphproof · cited by 2
- nhdsSet_nhdsKerproof · cited by 2
- nhdsSet_prod_leproof · cited by 2
- Disjoint.exists_uniform_thickeningproof · cited by 1
- IsCompact.lift'_closure_nhdsSetproof · cited by 1
- IsCompact.mem_nhdsSet_prod_of_forallproof · cited by 1
- exists_contMDiffMap_zero_one_nhds_of_isClosedproof · cited by 1
- upperHemicontinuousWithinAt_iff_preimage_Iicproof · cited by 1