English
The product of a charted space with itself is canonically the charted space on the product of the bases.
Русский
Произведение charted пространства с самим собой является канонически charted пространством на произведении оснований.
LaTeX
$$chartedSpaceSelf_prod$$
Lean4
/-- The product of two charted spaces is naturally a charted space, with the canonical
construction of the atlas of product maps. -/
instance prodChartedSpace (H : Type*) [TopologicalSpace H] (M : Type*) [TopologicalSpace M] [ChartedSpace H M]
(H' : Type*) [TopologicalSpace H'] (M' : Type*) [TopologicalSpace M'] [ChartedSpace H' M'] :
ChartedSpace (ModelProd H H') (M × M')
where
atlas := image2 OpenPartialHomeomorph.prod (atlas H M) (atlas H' M')
chartAt x := (chartAt H x.1).prod (chartAt H' x.2)
mem_chart_source x := ⟨mem_chart_source H x.1, mem_chart_source H' x.2⟩
chart_mem_atlas x := mem_image2_of_mem (chart_mem_atlas H x.1) (chart_mem_atlas H' x.2)
