EDB — 0W1

Exercises

1. [0W1]Reflect on the statements:

   • A closed set \(C\) inside a complete metric space \((X,d)\) is complete (when viewed as a metric space \((C,d)\)).

   • The set \(C=\{ 0\} ∪\{ 1/n : n∈ℕ\} \) is closed in \(ℝ\), so \(C\) is complete with distance \(d(x,y)=|x-y|\).

   • \(C\) is composed of countably many points.

   • A singleton \(\{ x\} \) is a closed set with an empty internal part.

   Why is there no contradiction?

   Solution 1

   [0W2]↺↻