continuous_map — Mathlib

English

The equivalence above holds for any X,Y with suitable topologies, reflecting the extension property of OnePoint.map.

Русский

Эквалентность вышеуказанного условия действует для любых X,Y с подходящими топологиями, отражая свойство расширения OnePoint.map.

LaTeX

$$$\\text{Continuous}(\\mathrm{OnePoint.map} f) \\iff \\bigl(\\text{Continuous}(f) \\wedge \\operatorname{Tendsto}(f, \\mathrm{coclosedCompact}(X), \\mathrm{coclosedCompact}(Y))\\bigr).$$$

Lean4

theorem continuous_map [TopologicalSpace Y] {f : X → Y} (hc : Continuous f)
    (h : Tendsto f (coclosedCompact X) (coclosedCompact Y)) : Continuous (OnePoint.map f) :=
  continuous_map_iff.mpr ⟨hc, h⟩

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