isCompact — Mathlib

English

In a Noetherian space, every subset is compact.

Русский

В пространстве Ноetherian каждый подмножество является компактным.

LaTeX

$$$\\forall s \\subseteq \\alpha,\\; IsCompact(s).$$$

Lean4

/-- In a Noetherian space, all sets are compact. -/
protected theorem isCompact [NoetherianSpace α] (s : Set α) : IsCompact s :=
  by
  refine isCompact_iff_finite_subcover.2 fun U hUo hs => ?_
  rcases
    ((noetherianSpace_iff_opens α).mp ‹_› ⟨⋃ i, U i, isOpen_iUnion hUo⟩).elim_finite_subcover U hUo Set.Subset.rfl with
    ⟨t, ht⟩
  exact ⟨t, hs.trans ht⟩

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