English
Given a closed set s in X that is compact, the set ((↑) '' s)ᶜ is open in OnePoint X; equivalently, the open sets of OnePoint X corresponding to closed-compact subsets of X are given as the complement of the image.
Русский
Для замкнутого компактного множества s ⊆ X, множество ((i)''s)ᶜ открыто в OnePoint X; открытые множества соответствуют замкнуто-компактным подмножствам X.
LaTeX
$$$s\\text{ closed in } X,\\; s\\text{ compact} \\Rightarrow ((\\iota)''s)^{c} \\text{ is open in } \\mathrm{OnePoint} X.$$$
Lean4
/-- An open set in `OnePoint X` constructed from a closed compact set in `X` -/
def opensOfCompl (s : Set X) (h₁ : IsClosed s) (h₂ : IsCompact s) : TopologicalSpace.Opens (OnePoint X) :=
⟨((↑) '' s)ᶜ, isOpen_compl_image_coe.2 ⟨h₁, h₂⟩⟩
