@@ -594,6 +594,10 @@ theorem isBounded_iff {s : Set α} :
594594 rw [isBounded_def, ← Filter.mem_sets, @PseudoMetricSpace.cobounded_sets α, mem_setOf_eq,
595595 compl_compl]
596596
597+ lemma boundedSpace_iff : BoundedSpace α ↔ ∃ C, ∀ a b : α, dist a b ≤ C := by
598+ rw [← isBounded_univ, Metric.isBounded_iff]
599+ simp
600+
597601theorem isBounded_iff_eventually {s : Set α} :
598602 IsBounded s ↔ ∀ᶠ C in atTop, ∀ ⦃x⦄, x ∈ s → ∀ ⦃y⦄, y ∈ s → dist x y ≤ C :=
599603 isBounded_iff.trans
@@ -610,6 +614,13 @@ theorem isBounded_iff_nndist {s : Set α} :
610614 simp only [isBounded_iff_exists_ge 0 , NNReal.exists, ← NNReal.coe_le_coe, ← dist_nndist,
611615 NNReal.coe_mk, exists_prop]
612616
617+ lemma boundedSpace_iff_nndist : BoundedSpace α ↔ ∃ C, ∀ a b : α, nndist a b ≤ C := by
618+ rw [← isBounded_univ, Metric.isBounded_iff_nndist]
619+ simp
620+
621+ lemma boundedSpace_iff_edist : BoundedSpace α ↔ ∃ C : ℝ≥0 , ∀ a b : α, edist a b ≤ C := by
622+ simp [Metric.boundedSpace_iff_nndist]
623+
613624theorem toUniformSpace_eq :
614625 ‹PseudoMetricSpace α›.toUniformSpace = .ofDist dist dist_self dist_comm dist_triangle :=
615626 UniformSpace.ext PseudoMetricSpace.uniformity_dist
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