continuous_curry — Mathlib

English

If a net F_i tends to f in the compact-open topology, then restricting to s preserves convergence: F_i|_s → f|_s.

Русский

Если сеть F_i сходится к f в компактно-открытой топологии, то ограничение на s сохраняет сходность: F_i|_s → f|_s.

LaTeX

$$$\text{Tendsto}(i\mapsto F(i)\restriction s)\ l\rightarrow\ NHDS(f\restriction s)$$$

Lean4

/-- The currying process is a continuous map between function spaces. -/
theorem continuous_curry [LocallyCompactSpace (X × Y)] : Continuous (curry : C(X × Y, Z) → C(X, C(Y, Z))) :=
  by
  apply continuous_of_continuous_uncurry
  apply continuous_of_continuous_uncurry
  rw [← (Homeomorph.prodAssoc _ _ _).symm.comp_continuous_iff']
  exact continuous_eval

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