HasBoundedInternalCoveringNumber #
def
HasBoundedInternalCoveringNumber
{T : Type u_1}
[PseudoEMetricSpace T]
(A : Set T)
(c : ENNReal)
(d : ℝ)
:
Equations
- HasBoundedInternalCoveringNumber A c d = ∀ ε ≤ EMetric.diam A, ↑(internalCoveringNumber ε A) ≤ c * ε⁻¹ ^ d
Instances For
theorem
HasBoundedInternalCoveringNumber.internalCoveringNumber_lt_top
{T : Type u_1}
[PseudoEMetricSpace T]
{A : Set T}
{c ε : ENNReal}
{d : ℝ}
(h : HasBoundedInternalCoveringNumber A c d)
(hε_ne : ε ≠ 0)
(hc : c ≠ ⊤)
(hd : 0 ≤ d)
:
theorem
HasBoundedInternalCoveringNumber.diam_lt_top
{T : Type u_1}
[PseudoEMetricSpace T]
{A : Set T}
{c : ENNReal}
{d : ℝ}
(h : HasBoundedInternalCoveringNumber A c d)
(hd : 0 < d)
:
theorem
HasBoundedInternalCoveringNumber.subset
{T : Type u_1}
[PseudoEMetricSpace T]
{A : Set T}
{c : ENNReal}
{d : ℝ}
{B : Set T}
(h : HasBoundedInternalCoveringNumber A c d)
(hBA : B ⊆ A)
(hd : 0 ≤ d)
:
HasBoundedInternalCoveringNumber B (2 ^ d * c) d
structure
IsCoverWithBoundedCoveringNumber
{T : Type u_1}
[PseudoEMetricSpace T]
(C : ℕ → Set T)
(A : Set T)
(c : ℕ → ENNReal)
(d : ℕ → ℝ)
:
- totallyBounded (n : ℕ) : TotallyBounded (C n)
- hasBoundedCoveringNumber (n : ℕ) : HasBoundedInternalCoveringNumber (C n) (c n) (d n)