toPullbackDiag — Mathlib

English

A function f is continuous iff it is both lower and upper semicontinuous on the whole space (univ).

Русский

Функция f непрерывна тогда и только тогда, когда она нижне и верхнеполупрерывна на всей пространстве (univ).

LaTeX

$$$$\\operatorname{Continuous} f \\iff \\operatorname{LowerSemicontinuous} f \\land \\operatorname{UpperSemicontinuous} f.$$$$

Lean4

protected theorem toPullbackDiag (f : X → Y) : IsEmbedding (toPullbackDiag f) :=
  .mk' _ (injective_toPullbackDiag f) fun x ↦ by
    simp [nhds_induced, Filter.comap_comap, nhds_prod_eq, Filter.comap_prod, Function.comp_def, Filter.comap_id']

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