tendsto_cocompact — Mathlib

English

A closed embedding is proper: the preimage of cocompact sets is cocompact.

Русский

Замкнутое вложение является правильным: предобраз кокофактов остаётся кокофактом.

LaTeX

$$Tendsto f (Filter.cocompact X) (Filter.cocompact Y)$$

Lean4

/-- A closed embedding is proper, i.e., inverse images of compact sets are contained in compacts.
Moreover, the preimage of a compact set is compact, see `IsClosedEmbedding.isCompact_preimage`. -/
theorem tendsto_cocompact (hf : IsClosedEmbedding f) : Tendsto f (Filter.cocompact X) (Filter.cocompact Y) :=
  Filter.hasBasis_cocompact.tendsto_right_iff.mpr fun _K hK => (hf.isCompact_preimage hK).compl_mem_cocompact

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