CRN 10459: HW 4

Problem 16. Show: If $X\subseteq \mathbb{R}$ is both open and closed, then $X=\mathbb{R}$ or $X=\emptyset$.

Problem 17. Exercise 3.2.7.

Problem 18. Exercise 3.2.14.

Problem 19. Consider the following sets: \[A=\left\{1,\frac{1}{2},\frac{1}{3},\frac{1}{4}\ldots\right\},\quad B=\left\{1,\frac{1}{2},\frac{2}{3},\frac{3}{4},\frac{4}{5}\ldots\right\}, \quad C=\mathbb{Q}\cap[0,1]\] For the sets that are compact, explain why. For the other ones, show that they have an open cover without finite subcover.

Problem 20. Exercise 3.4.6.