English
For a continuous map f, the comap of the cocompact filter on Y is less than or equal to the cocompact filter on X.
Русский
При непрерывном отображении f сопряжение кокофичество на Y не умножает кокофактивность на X.
LaTeX
$$$(\\\\mathrm{cocompact}(Y)).comap\, f \\\\le \\\\mathrm{cocompact}(X)$$$
Lean4
/-- The comap of the cocompact filter on `Y` by a continuous function `f : X → Y` is less than or
equal to the cocompact filter on `X`.
This is a reformulation of the fact that images of compact sets are compact. -/
theorem comap_cocompact_le {f : X → Y} (hf : Continuous f) : (Filter.cocompact Y).comap f ≤ Filter.cocompact X :=
by
rw [(Filter.hasBasis_cocompact.comap f).le_basis_iff Filter.hasBasis_cocompact]
intro t ht
refine ⟨f '' t, ht.image hf, ?_⟩
simpa using t.subset_preimage_image f
