Theorems · Theorem · general topology
Metric.Sigma.isOpen_iff
∀ {ι : Type u_1} {E : ι → Type u_2} [inst : (i : ι) → MetricSpace (E i)] (s : Set ((i : ι) × E i)),
IsOpen s ↔ ∀ x ∈ s, ∃ ε > 0, ∀ (y : (i : ι) × E i), dist x y < ε → y ∈ s- Defined in
- Mathlib.Topology.MetricSpace.Gluing
- Cited by
- 0 results in Mathlib
- Foundations
- Depth 118 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- MetricSpace
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites18
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
- Set.preimageproof · cited by 4,946
- IsOpenstatement and proof · cited by 2,400
- MetricSpacestatement and proof · cited by 1,684
- Dist.diststatement and proof · cited by 1,539
- eq_or_neproof · cited by 1,117
- Metric.ballproof · cited by 735
- LT.lt.trans_leproof · cited by 678
- zero_lt_oneproof · cited by 598
- lt_irreflproof · cited by 190
- min_le_rightproof · cited by 88
Cited by1
Results whose statement or proof uses this declaration.
- Metric.Sigma.metricSpaceproof · cited by 2