Theorems · Theorem · functional analysis
frontier_ball
∀ {E : Type u_1} [inst : SeminormedAddCommGroup E] [NormedSpace ℝ E] (x : E) {r : ℝ},
r ≠ 0 → frontier (Metric.ball x r) = Metric.sphere x r- Cited by
- 1 results in Mathlib
- Foundations
- Depth 164 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites13
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
- NormedSpacestatement and proof · cited by 12,499
- SeminormedAddCommGroupstatement and proof · cited by 2,671
- Metric.ballstatement and proof · cited by 735
- interiorproof · cited by 714
- Metric.closedBallproof · cited by 704
- Metric.spherestatement and proof · cited by 371
- frontierstatement · cited by 214
- Metric.isOpen_ballproof · cited by 63
- IsOpen.interior_eqproof · cited by 58
- closure_ballproof · cited by 20
Cited by1
Results whose statement or proof uses this declaration.
- exists_homeomorph_image_interior_closure_frontier_eq_unitBallproof · cited by 0