English
There exists a SupBotHom structure on coverEntropy with mapping to the real extended line, capturing monotone set relations via entropy.
Русский
Существует структура SupBotHom на coverEntropy, отображающая множество в зависимостях энтропии и сохраняющая монотонность.
LaTeX
$$$\\text{coverEntropy\_supBotHom }(T) :\\ SupBotHom (Set X) \\mathbb{R}_{\\ge -\\infty} $ with $\\text{toFun} = \\text{coverEntropy}(T)$, etc.$$
Lean4
/-- Topological entropy `CoverEntropy T` as a `SupBotHom` function of the subset. -/
noncomputable def coverEntropy_supBotHom (T : X → X) : SupBotHom (Set X) EReal
where
toFun := coverEntropy T
map_sup' := fun _ _ ↦ coverEntropy_union
map_bot' := coverEntropy_empty