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 $A$ of $X$ has a limit point belonging to $X$. |
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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. | ||
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| + | '''Problem 30.''' Show the following: Let $O$ be a non-empty open set of real numbers. Then there are open intervals $I_n$, $n\in \mathbb{N}$, such that (1) $\displaystyle O=\bigcup_{n\in\mathbb{N}} I_n$, and (2) $I_m\cap I_n=\emptyset$ for all $m\neq n$. | ||
Revision as of 19:23, 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 $A$ 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 the following: Let $O$ be a non-empty open set of real numbers. Then there are open intervals $I_n$, $n\in \mathbb{N}$, such that (1) $\displaystyle O=\bigcup_{n\in\mathbb{N}} I_n$, and (2) $I_m\cap I_n=\emptyset$ for all $m\neq n$.
