English
A chart on a discrete space is constant; the chartAt is constant for each x.
Русский
На дискретном пространстве карта постоянная; chartAt constant for every x.
LaTeX
$$chartedSpace_of_discreteTopology_chartAt$$
Lean4
/-- Any discrete space is a charted space over a singleton set.
We keep this as a definition (not an instance) to avoid instance search trying to search for
`DiscreteTopology` or `Unique` instances.
-/
def of_discreteTopology [TopologicalSpace M] [TopologicalSpace H] [DiscreteTopology M] [h : Unique H] : ChartedSpace H M
where
atlas :=
letI f := fun x : M ↦ OpenPartialHomeomorph.const (isOpen_discrete { x }) (isOpen_discrete { h.default })
Set.image f univ
chartAt x := OpenPartialHomeomorph.const (isOpen_discrete { x }) (isOpen_discrete { h.default })
mem_chart_source x := by simp
chart_mem_atlas x := by simp
