English
Let E be a normed group and e: X → E a closed embedding with X having a discrete topology, where X is a discrete space and E is a proper space. Then the norm of e(x) tends to infinity as x runs off to infinity in X; equivalently, the function x ↦ ∥e(x)∥ diverges along the cofinite filter.
Русский
Пусть e: X → E — замыкованное внедрение, где X дискретно топологически, и E — правильное пространство. Тогда при alejening к бесконечности ||e(x)|| стремится к бесконечности.
LaTeX
$$$\operatorname{Tendsto}(\|e\| \circ e)\;\mathrm{cofinite}\;\mathrm{atTop}$$$
Lean4
@[to_additive tendsto_norm_comp_cofinite_atTop_of_isClosedEmbedding]
theorem tendsto_norm_comp_cofinite_atTop_of_isClosedEmbedding' {X : Type*} [TopologicalSpace X] [DiscreteTopology X]
[ProperSpace E] {e : X → E} (he : Topology.IsClosedEmbedding e) : Tendsto (norm ∘ e) cofinite atTop :=
by
rw [← Filter.cocompact_eq_cofinite X]
apply tendsto_norm_cocompact_atTop'.comp (Topology.IsClosedEmbedding.tendsto_cocompact he)
