compactlySupported_eq_top_iff — Mathlib

English

For nontrivial γ, C_cb(α, γ) = ⊤ if and only if IsCompact (Set.univ : Set α).

Русский

Для ненулевого γ, C_cb(α, γ) = ⊤ тогда и только если IsCompact (множество-где-едино).

LaTeX

$$$C_{cb}(\\alpha, \\gamma) = \\top \\iff IsCompact(\\mathrm{Set.univ})$$$

Lean4

theorem compactlySupported_eq_top_iff [Nontrivial γ] : C_cb(α, γ) = ⊤ ↔ IsCompact (Set.univ : Set α) :=
  by
  refine ⟨fun h ↦ ?_, compactlySupported_eq_top_of_isCompact⟩
  obtain ⟨x, hx⟩ := exists_ne (0 : γ)
  simpa [tsupport, Function.support_const hx] using (mem_compactlySupported (f := const α x).mp (by simp [h])).isCompact

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