10 Dimension[0YH]

Let \((X,d)\) be a metric space. Let in the following \(K\) a compact non-empty subset of \(X\), and \(N(𝜌)\) the minimum number of balls of radius \(𝜌\) that are needed to cover \(K\).  ¹

Note that this definition depends a priori on the choice of the distance, i.e. \(N=N(𝜌,K,d)\) and \(\dim =\dim (K,d)\). See in particular [0YZ]↺↻.

Exercises

1. [0YK]↺↻

2. [0YN]↺↻

3. [0YQ]↺↻

4. [0YS]↺↻

5. [0YV]↺↻

6. [0YX]↺↻

7. [0YZ]↺↻

8. [0Z1]↺↻

9. [0Z3]↺↻

10. [0Z7]↺↻

11. [0ZB]↺↻

12. [0ZD]↺↻

13. [0ZG]↺↻

14. [0ZJ]↺↻

15. [0ZM]↺↻

16. [0ZP]↺↻

17. [0ZR]↺↻