English
In a Polish space, every closed set is clopenable: there exists a Polish topology refining the original in which the set is both open and closed.
Русский
В полиспейсе каждое замкнутое множество допускает полис topology для которого множество открыто и закрыто.
LaTeX
$$$\exists t' : TopologicalSpace\;\alpha,\ t' \le t \land \text{PolishSpace }(\alpha,t') \land \text{IsClosed}[t'](s) \land \text{IsOpen}[t'](s)$$$
Lean4
/-- A set in a topological space is clopenable if there exists a finer Polish topology for which
this set is open and closed. It turns out that this notion is equivalent to being Borel-measurable,
but this is nontrivial (see `isClopenable_iff_measurableSet`). -/
def IsClopenable [t : TopologicalSpace α] (s : Set α) : Prop :=
∃ t' : TopologicalSpace α, t' ≤ t ∧ @PolishSpace α t' ∧ IsClosed[t'] s ∧ IsOpen[t'] s
