Theorems · Theorem · functional analysis
closure_ball
∀ {E : Type u_1} [inst : SeminormedAddCommGroup E] [NormedSpace ℝ E] (x : E) {r : ℝ},
r ≠ 0 → closure (Metric.ball x r) = Metric.closedBall x r- Cited by
- 20 results in Mathlib
- Foundations
- Depth 163 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites39
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement and proof · cited by 53,352
- Realstatement and proof · cited by 25,697
- TopologicalSpaceproof · cited by 24,529
- NormedSpacestatement and proof · cited by 12,499
- LinearOrderproof · cited by 8,572
- Norm.normproof · cited by 5,413
- mul_oneproof · cited by 3,885
- LE.le.transproof · cited by 3,151
- SeminormedAddCommGroupstatement and proof · cited by 2,671
- mul_commproof · cited by 2,262
- one_smulproof · cited by 1,374
- OrderTopologyproof · cited by 1,355
Cited by20
Results whose statement or proof uses this declaration.
- DiffContOnCl.continuousOn_ballproof · cited by 6
- Topology.RelCWComplex.closure_openCell_eq_closedCellproof · cited by 3
- DiffContOnCl.circleAverage_smul_divproof · cited by 2
- ExistsContDiffBumpBase.w_compact_supportproof · cited by 2
- AnalyticAt.eventually_constant_or_nhds_le_map_nhds_auxproof · cited by 1
- ContDiffBump.tsupport_eqproof · cited by 1
- ContDiffBump.tsupport_normed_eqproof · cited by 1
- Complex.ECanonicalDecomp.eq_smul_meromorphicTrailingCoeffAtproof · cited by 1
- Metric.closedBall_infDist_compl_subset_closureproof · cited by 1
- ExistsContDiffBumpBase.u_compact_supportproof · cited by 1
- Euclidean.closure_ballproof · cited by 1
- DiffContOnCl.two_pi_i_inv_smul_circleIntegral_sub_inv_smulproof · cited by 1