English
The zero module has the module topology; if A is a subsingleton, IsModuleTopology R A holds with eq_moduleTopology' given by Subsingleton.elim.
Русский
Нулевой модуль имеет модульную топологию; если A — подсобственный множество, выполняется IsModuleTopology R A и eq_moduleTopology' задаётся через Subsingleton.elim.
LaTeX
$$$[\text{Subsingleton } A] \Rightarrow \text{IsModuleTopology } R A \text{ with } eq_moduleTopology' := \text{Subsingleton.elim } _ _ $$$
Lean4
/-- If `A` is a topological `R`-module and the identity map from (`A` with its given
topology) to (`A` with the module topology) is continuous, then the topology on `A` is
the module topology. -/
theorem of_continuous_id [ContinuousAdd A] [ContinuousSMul R A] (h : @Continuous A A τA (moduleTopology R A) id) :
IsModuleTopology R A where
-- The topologies are equal because each is finer than the other. One inclusion
-- follows from the continuity hypothesis; the other is because the module topology
-- is the inf of all the topologies making `A` a topological module.
eq_moduleTopology' := le_antisymm (continuous_id_iff_le.1 h) (moduleTopology_le _ _)
