close.scala

final case class Epsilon[A](value: A)
object Epsilon {
  def apply[A: Field](x: Double): A = Field[A].fromDouble(x)
}

// This can be part of the MetricSpace ops.

implicit class ApproxEq[A](x: A) {
  def =~=[B](y: A)(implicit ms: MetricSpace[A, B], e: Epsilon[B], o: Order[B]): Boolean =
    MetricSpace.close(x, y)(ms, e, o)
}

// In algebra/MetricSpace.scala

object MetricSpace {
  def close[A, B](x: A, y: A)(implicit ms: MetricSpace[A, B], e: Epsilon[B], o: Order[B]): Boolean = {
    ms.distance(x, y) < e.value
  }

  implicit def trivialMetricSpace[A: IsReal: Rng] = new MetricSpace[A, A] {
    def distance(x: A, y: A): A = (x - y).abs
  }
}

Examples:

implicit val e = Epsilon(1)
"Thomas" =~= "thomas"

implicit val e2 = Epsilon(0.00001)
Vector(1.0, 1.0) =~= Vector(1.00000001, 0.99999999)

I think we'd actually want to somehow tie Epsilon to both types in the MEtricSpace too....