Theorems · Theorem · dynamical systems
eq_empty_or_univ_of_smul_invariant_closed
∀ (M : Type u_1) {α : Type u_3} [inst : Monoid M] [inst_1 : TopologicalSpace α] [inst_2 : MulAction M α]
[MulAction.IsMinimal M α] {s : Set α}, IsClosed s → (∀ (c : M), c • s ⊆ s) → s = ∅ ∨ s = Set.univ- Defined in
- Mathlib.Dynamics.Minimal
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 69 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites13
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.univstatement · cited by 3,945
- Monoidstatement and proof · cited by 3,887
- Set.Nonemptyproof · cited by 2,627
- IsClosedstatement and proof · cited by 1,639
- MulActionstatement and proof · cited by 1,294
- Set.smulSetstatement · cited by 608
- Set.eq_empty_or_nonemptyproof · cited by 248
- IsClosed.closure_eqproof · cited by 139
- Dense.closure_eqproof · cited by 24
- MulAction.IsMinimalstatement and proof · cited by 16
Cited by1
Results whose statement or proof uses this declaration.
- isMinimal_iff_isClosed_smul_invariantproof · cited by 0