CRN 11982: HW 6
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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: Recall that every bounded infinite set has a limit point. | + | * 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 | '''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 | * $\displaystyle O=\bigcup_{n\in\mathbb{N}} I_n$, and | ||
* for all $m,n\in\mathbb{N}$: $I_m\cap I_n=\emptyset$ or $I_m=I_n$. | * for all $m,n\in\mathbb{N}$: $I_m\cap I_n=\emptyset$ or $I_m=I_n$. | ||
Latest revision as of 20:18, 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 $I_m=I_n$.
