Theorems · Theorem · general topology
EMetric.mem_closure_iff
∀ {α : Type u} [inst : PseudoEMetricSpace α] {x : α} {s : Set α}, x ∈ closure s ↔ ∀ ε > 0, ∃ y ∈ s, edist x y < εε-characterization of the closure in pseudoemetric spaces
- Defined in
- Mathlib.Topology.EMetricSpace.Defs
- Cited by
- 6 results in Mathlib
- Foundations
- Depth 141 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- PseudoEMetricSpace
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites8
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
- ENNRealstatement and proof · cited by 9,879
- PseudoEMetricSpacestatement and proof · cited by 1,536
- closurestatement · cited by 1,254
- EDist.ediststatement and proof · cited by 735
- PseudoEMetricSpace.edist_commproof · cited by 52
- mem_closure_iff_nhds_basisproof · cited by 13
- Metric.nhds_basis_eballproof · cited by 10
Cited by6
Results whose statement or proof uses this declaration.
- Metric.infEDist_closureproof · cited by 8
- continuousOn_prod_of_subset_closure_continuousOn_lipschitzOnWithproof · cited by 3
- Metric.mem_closure_iff_infEDist_zeroproof · cited by 3
- AnalyticOnNhd.eqOn_zero_of_preconnected_of_eventuallyEq_zero_auxproof · cited by 1
- MeasureTheory.SimpleFunc.tendsto_nearestPtproof · cited by 1
- VitaliFamily.ae_tendsto_lintegral_enorm_sub_div'_of_integrableproof · cited by 1