EDB — 0Q0

Exercises

1. [0Q0]Prerequisites:[0Q3]↺↻,[0P3]↺↻,[0PY]↺↻, [0PN]↺↻.Let \(r{\gt}0\).

   Let \(D(x,r){\stackrel{.}{=}}\{ y∈ X : d(x,y)≤ r\} \) be the disk; show that \(\overline{B(x,r)}⊆ D(x,r)\) and that \(B(x,r)⊆ {{D(x,r)}^\circ }\).

   Let \(S(x,r){\stackrel{.}{=}}\{ y∈ X : d(x,y)= r\} \) be the sphere; show that \(∂{B(x,r)}⊆ S(x,r)\).

   Find examples of metric spaces in which the above equalities (one, or both) do not hold.

   Find an example of a metric space where there is a disk that is open ¹ .

   (See also [0SM]↺↻ for the case of space \(ℝ^ n\)).

   Solution 1

   [0Q1]↺↻