toMetricSpace — Mathlib

English

If a set has pairwise ε-separated points for ε>0, then it is closed.

Русский

Если множество имеет пары точек, отделённых на расстоянии не менее ε>0, то множество замкнуто.

LaTeX

$$$ hε>0, hs.Pairwise(ε ≤ dist(x,y)) \Rightarrow IsClosed(s) $$$

Lean4

/-- One gets a metric space from an emetric space if the edistance
is everywhere finite, by pushing the edistance to reals. We set it up so that the edist and the
uniformity are defeq in the metric space and the emetric space. -/
def toMetricSpace {α : Type u} [EMetricSpace α] (h : ∀ x y : α, edist x y ≠ ⊤) : MetricSpace α :=
  EMetricSpace.toMetricSpaceOfDist (fun x y => ENNReal.toReal (edist x y)) h fun _ _ => rfl

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