CRN 11982: HW 6
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'''Problem 26.''' Exercise 3.3.1 | '''Problem 26.''' Exercise 3.3.1 | ||
− | '''Problem 27.''' Exercise 3.3. | + | '''Problem 27.''' Exercise 3.3.9 (b,e) |
− | '''Problem 28.''' Exercise 3.3. | + | '''Problem 28.''' Exercise 3.3.10 |
− | '''Problem 29 | + | '''Problem 29.''' A set $X$ is called ''LP-compact'', if every infinite subset of $X$ has a limit point belonging to $X$. |
− | + | ||
− | + | ||
Show that a set is LP-compact if and only if it is closed and bounded. | Show that a set is LP-compact if and only if it is closed and bounded. | ||
− | Hint: | + | * Hint: Recall that every bounded infinite set has a limit point. |
+ | |||
+ | '''Problem 30.''' Show that every non-empty open set is the countable union of pairwise disjoint open intervals: Let $O$ be a non-empty open set of real numbers. Then there are open intervals $I_n$, $n\in \mathbb{N}$, such that | ||
+ | * $\displaystyle O=\bigcup_{n\in\mathbb{N}} I_n$, and | ||
+ | * for all $m,n\in\mathbb{N}$: $I_m\cap I_n=\emptyset$ or $I_m=I_n$. |
Latest revision as of 21:18, 28 October 2014
Problem 26. Exercise 3.3.1
Problem 27. Exercise 3.3.9 (b,e)
Problem 28. Exercise 3.3.10
Problem 29. A set $X$ is called LP-compact, if every infinite subset of $X$ has a limit point belonging to $X$.
Show that a set is LP-compact if and only if it is closed and bounded.
- Hint: Recall that every bounded infinite set has a limit point.
Problem 30. Show that every non-empty open set is the countable union of pairwise disjoint open intervals: Let $O$ be a non-empty open set of real numbers. Then there are open intervals $I_n$, $n\in \mathbb{N}$, such that
- $\displaystyle O=\bigcup_{n\in\mathbb{N}} I_n$, and
- for all $m,n\in\mathbb{N}$: $I_m\cap I_n=\emptyset$ or $I_m=I_n$.