Theorems · Theorem · logic and foundations
IsClub.dirSupClosed
∀ {α : Type u_1} [inst : LinearOrder α] {s : Set α}, IsClub s → DirSupClosed sClub sets are closed under suprema. If α is a well-order with the order topology, this
condition is equivalent to IsClosed s.
- Cited by
- 4 results in Mathlib
- Foundations
- Depth 17 from the axioms · uses propext
- Assumes
- LinearOrder
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites4
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
- LinearOrderstatement and proof · cited by 8,572
- DirSupClosedstatement · cited by 31
- IsClubstatement and proof · cited by 30
Cited by4
Results whose statement or proof uses this declaration.
- IsClub.sInterproof · cited by 2
- IsClub.sInter_of_orderTopproof · cited by 2
- IsClub.isLUB_memproof · cited by 2
- IsClub.unionproof · cited by 0