Homework 10: Metric Spaces

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

1. Section 3.6, #7.
2. For \(x,y\in \mathbb{R}\), define \begin{align} d_1(x,y)&=|x|+|y|\\ d_2(x,y)&=(x-y)^2\\ d_3(x,y)&=|x-2y|\\ d_4(x,y)&=|x^2-y^2| \end{align} Determine whether each of these is a metric on \( \mathbb{R}\). Justify your answers.
3. Let \(X\) be a nonempty set and let \(d\) be the discrete metric on \(X\). Prove that every subset of \(X\) is both open and closed.
4. Consider \(\mathbb{R}\) with the discrete metric. Prove that \(E=[0,1]\) is closed and bounded in \(\mathbb{R}\), but not compact. (Note that closed, bounded, and compact are in reference to the discrete metric.)
5. Consider \(\mathbb{Q}\), viewed as a metric subspace of \(\mathbb{R}\) with Euclidean metric. Let \(E\) be the subset \(\{p\in \mathbb{Q}: 2\lt p^2\lt 3\}\) of \(\mathbb{Q}\). Prove that in this metric subspace, \(E\) is closed and bounded, but not compact.
6. (GRE Mathematics Subject Test) Let \(d\) be a metric on a set \(X\). Which of the following is also a metric on \(X\)?
   1. \(4+d\)
   2. \(e^d-1\)
   3. \(d-|d|\)
   4. \(d^2\)
   5. \(\sqrt{d}\)