Exercises

1. Let \((M,d)\) be a metric space, and let \(A\subseteq M\).

  1. Prove that \(A'\) is closed.
  2. Prove that \(A\) and \(\overline A\) have the same limit points.
  3. Prove or disprove that \(A\) and \(A'\) have the same limit points.

2. Let \((M,d)\) be a metric space and let \(A\subseteq M\). Prove that

  1. the set \(A\) is open if and only if \(A\cap\partial A=\varnothing\);
  2. the set \(A\) is closed if and only if \(\partial A\subseteq A\);
  3. the set \(A\) is dense in \(M\) if and only if \(\operatorname{ext} A = \varnothing\).

Solution. (1) Suppose \(A\) is open. Since \(\operatorname{int} A\) and \(\partial A\) is disjoint, \(A\subseteq \operatorname{int} A\) implies \(A\cap \partial A = \varnothing.\) Conversely, suppose \(A\cap \partial A = \varnothing.\) Then \(x\in A\) implies \(x\notin \partial A,\) so that \(x\in \operatorname{int} A\) or \(x\in \operatorname{ext} A.\) Since \(A\) and \(\operatorname{ext} A\) are disjoint, \(x\in \operatorname{int} A.\) Hence \(A\subseteq \operatorname{int} A,\) so \(A\) is open. (2) If \(\partial A\subseteq A,\) then \(\operatorname{int} A \cup \partial A \subseteq A.\) Hence \(A'\subseteq \overline A \subseteq A.\) Thus \(A\) is closed. Conversely, if \(A'\subseteq A\) then \(\overline A\subseteq A.\) Hence \(\partial A \subseteq \overline A \subseteq A.\) (3) Since \(M\subseteq M\setminus \operatorname{ext} A\) is equivalent to \(\operatorname{ext} A=\varnothing,\) if \(A\) is dense in \(M\) then \(\operatorname{ext} A=\varnothing,\) and vice versa.\(\square\)

3. Let \(M\) be a metric space, and let \(A_1,A_2,\ldots\subseteq M\). Prove that

  1. if \(B_n = \bigcup_{i=1}^n A_i\), then \(\overline {B_n} = \bigcup_{i=1}^n \overline {A_i}\);
  2. if \(B = \bigcup_{i=1}^{\infty} A_i\), then \(\bigcup_{i=1}^{\infty}\overline {A_i}\subseteq \overline B\).

4. Let \(K\subseteq \mathbb R\) consist of \(0\) and the numbers \(1,1/2,1/3,\ldots.\) Prove that \(K\) is compact without using the Heine–Borel theorem.