locallyCompactSpace_iff — Mathlib

English

The codomain and domain of a homeomorphism are locally compact simultaneously: LocallyCompactSpace X iff LocallyCompactSpace Y.

Русский

Область и областьюCode две локально компактны одновременно: LocallyCompactSpace X эквивалентно LocallyCompactSpace Y.

LaTeX

$$$LocallyCompactSpace X \\iff LocallyCompactSpace Y$$$

Lean4

/-- The codomain of a homeomorphism is a locally compact space if and only if
the domain is a locally compact space. -/
theorem locallyCompactSpace_iff (h : X ≃ₜ Y) : LocallyCompactSpace X ↔ LocallyCompactSpace Y := by
  exact ⟨fun _ => h.symm.isOpenEmbedding.locallyCompactSpace, fun _ => h.isClosedEmbedding.locallyCompactSpace⟩

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