Theorems · Theorem · Lie groups

Topology.IsInducing.continuousSMul

∀ {M : Type u_1} {X : Type u_2} {Y : Type u_3} [inst : TopologicalSpace M] [inst_1 : TopologicalSpace X]
  [inst_2 : TopologicalSpace Y] [inst_3 : SMul M X] [ContinuousSMul M X] {g : Y → X} {N : Type u_5} [inst_5 : SMul N Y]
  [inst_6 : TopologicalSpace N] {f : N → M},
  Topology.IsInducing g → Continuous f → (∀ {c : N} {x : Y}, g (c • x) = f c • g x) → ContinuousSMul N Y

Suppose that N acts on X and M continuously acts on Y. Suppose that g : Y → X is an action homomorphism in the following sense: there exists a continuous function f : N → M such that g (c • x) = f c • g x. Then the action of N on X is continuous as well. In many cases, f = id so that g is an action homomorphism in the sense of MulActionHom. However, this version also works for semilinear maps and f = Units.val.

Defined in

Mathlib.Topology.Algebra.MulAction

Cited by

1 results in Mathlib

Foundations

Depth 76 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

TopologicalSpaceTopologicalSpaceTopologicalSpaceSMulContinuousSMulSMulTopologicalSpace