English
Let X, Y, Z be topological spaces and g: C(Y, Z) be a map that induces the topology on Y from Z. Then the postcomposition map (g ∘ −): C(X, Y) → C(X, Z), sending f to g ∘ f, is inducing (i.e., the compact-open topology on C(X, Z) is the pullback of the topology along this map).
Русский
Пусть X, Y, Z — топологические пространства, и g: C(Y, Z) задаёт топологию на Y как подпространство Z. Тогда отображение посткомпозиции (g ∘ −): C(X, Y) → C(X, Z), définée как f ↦ g ∘ f, индуктивно вместе с компактно-открытой топологией на C(X, Z).
LaTeX
$$$\\operatorname{IsInducing}(f \\mapsto g \\circ f)$$$
Lean4
/-- If `g : C(Y, Z)` is a topology inducing map,
then the composition `ContinuousMap.comp g : C(X, Y) → C(X, Z)` is a topology inducing map too. -/
theorem isInducing_postcomp (g : C(Y, Z)) (hg : IsInducing g) : IsInducing (g.comp : C(X, Y) → C(X, Z)) where
eq_induced := by
simp only [compactOpen_eq, induced_generateFrom_eq, image_image2, hg.setOf_isOpen, image2_image_right, MapsTo,
mem_preimage, preimage_setOf_eq, comp_apply]
