CRN 10459: HW 4
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HelmutKnaust (Talk | contribs) (Created page with "'''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.''' Exercis...") |
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'''Problem 17.''' Exercise 3.2.7. | '''Problem 17.''' Exercise 3.2.7. | ||
| − | '''Problem 18.''' Exercise 3.2. | + | '''Problem 18.''' Exercise 3.2.15. |
'''Problem 19.''' Consider the following sets: | '''Problem 19.''' Consider the following sets: | ||
Latest revision as of 10:48, 14 October 2025
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 3.2.15.
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.
