Theorems · Theorem · Lie groups
Submonoid.isOpen_units
∀ {M : Type u_1} [inst : TopologicalSpace M] [inst_1 : Monoid M] {U : Submonoid M}, IsOpen ↑U → IsOpen ↑U.unitsIf a submonoid is open in a topological monoid, then its units form an open subset of the units of the monoid.
- Defined in
- Mathlib.Topology.Algebra.Group.Units
- Cited by
- 0 results in Mathlib
- Foundations
- Depth 77 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- TopologicalSpaceMonoid
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Cites12
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- TopologicalSpacestatement and proof · cited by 24,529
- SetLike.coestatement and proof · cited by 8,199
- Monoidstatement and proof · cited by 3,887
- Subgroupstatement · cited by 3,593
- Submonoidstatement and proof · cited by 3,086
- Unitsstatement · cited by 2,804
- IsOpenstatement and proof · cited by 2,400
- IsOpen.preimageproof · cited by 147
- IsOpen.interproof · cited by 98
- Submonoid.unitsstatement · cited by 40
- Units.continuous_valproof · cited by 8
- Units.continuous_coe_invproof · cited by 4
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