Theorems · Theorem · general topology
ProperSpace.isCompact_closedBall
∀ {α : Type u} {inst : PseudoMetricSpace α} [self : ProperSpace α] (x : α) (r : ℝ), IsCompact (Metric.closedBall x r)- Defined in
- Mathlib.Topology.MetricSpace.ProperSpace
- Cited by
- 40 results in Mathlib
- Foundations
- Depth 96 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- ProperSpace
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites5
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Realstatement · cited by 25,697
- PseudoMetricSpacestatement and proof · cited by 1,550
- IsCompactstatement · cited by 1,282
- Metric.closedBallstatement · cited by 704
- ProperSpacestatement and proof · cited by 190
Cited by40
Results whose statement or proof uses this declaration.
- isCompact_sphereproof · cited by 11
- Metric.cobounded_eq_cocompactproof · cited by 8
- Metric.isCompact_of_isClosed_isBoundedproof · cited by 8
- InnerProductSpace.HarmonicOnNhd.circleAverage_eqproof · cited by 5
- MeasureTheory.addHaar_image_le_mul_of_det_ltproof · cited by 4
- upperHemicontinuous_spectrumproof · cited by 4
- exists_pos_lt_subset_ballproof · cited by 4
- Besicovitch.exists_goodδproof · cited by 3
- MeromorphicOn.circleAverage_log_normproof · cited by 3
- MeromorphicOn.divisor_ball_support_finiteproof · cited by 3
- Metric.mem_cocompact_of_closedBall_compl_subsetproof · cited by 3