Homework 8: Topology of \(\mathbb{R}\)

Instructions. Assignments should be stapled and written clearly and legibly. Problem 6 is optional.

1. Define the following (as in class): open set, closed set, limit point, \(A'\), \(\overline{A}\). In the problems below, you may use only these definitions. You may not use any theorems from the sections we've covered on topology.
2. Section 3.4, #14, 15.
3. Let \(A\subseteq B\) be two sets of real numbers. We say that \(A\) is dense in \(B\) if \(\overline{A}=B\). Prove that \(\mathbb{Q}\) is dense in \(\mathbb{R}\). (In other words, prove \(\overline{\mathbb{Q}}=\mathbb{R}\).)
4. Let \(A\) be a set of real numbers. Prove the following:
   1. \(A'\) is closed.
   2. \((\overline{A})'=A'\).
   3. \(\overline{A}\) is closed.
5. Let \(A\) be a set of real numbers. Prove that if \(B\) is a closed set and \(B\supseteq A\), then \(B\supseteq \overline{A}\). (Note. Problems 4(c) and 5 tell us that \(\overline{A}\) is the smallest closed set containing \(A\).)
6. (Challenge) For sets \(A,B \subseteq \mathbb{R}\), the Minkowski sum of \(A\) and \(B\) is defined to be \[ A+B=\{a+b: a\in A, b\in B\}.\]
   1. Prove that if \(A\) and \(B\) are open, then \(A+B\) is open.
   2. Disprove the following by giving a counterexample: If \(A\) and \(B\) are closed, then \(A+B\) is closed. (Hint: for any such counterexample, neither \(A\) nor \(B\) can be bounded.)