Theorems · Theorem · general topology
OnePoint.isOpen_iff_of_mem
∀ {X : Type u_1} [inst : TopologicalSpace X] {s : Set (OnePoint X)},
OnePoint.infty ∈ s → (IsOpen s ↔ IsClosed (OnePoint.some ⁻¹' s)ᶜ ∧ IsCompact (OnePoint.some ⁻¹' s)ᶜ)- Cited by
- 3 results in Mathlib
- Foundations
- Depth 90 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.
Cites11
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.preimagestatement and proof · cited by 4,946
- Compl.complstatement and proof · cited by 2,925
- IsOpenstatement and proof · cited by 2,400
- IsClosedstatement · cited by 1,639
- IsCompactstatement and proof · cited by 1,282
- OnePointstatement and proof · cited by 126
- OnePoint.inftystatement and proof · cited by 102
- OnePoint.somestatement and proof · cited by 76
- OnePoint.isOpen_iff_of_mem'proof · cited by 1
Cited by3
Results whose statement or proof uses this declaration.
- OnePoint.isOpen_compl_image_coeproof · cited by 2
- OnePoint.nhdsNE_infty_eqproof · cited by 1
- OnePoint.isClosed_iff_of_notMemproof · cited by 0