CRN 12109: HW 5
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| − | + | '''Problem 21.''' Exercise 3.2.2 (a-d) | |
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| + | '''Problem 22.''' Given a set $X$ of real numbers, let $L$ be the set of all limit points of $X$. Show that $L$ is closed. | ||
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| + | '''Problem 23.''' Show: If $X$ is both open and closed, then $X={\mathbb R}$ or $X=\emptyset$. | ||
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| + | '''Problem 24.''' Exercise 3.2.14 | ||
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| + | '''Problem 25.''' 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. | ||
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| + | Hint: Show first that every bounded infinite set has a limit point. | ||
Latest revision as of 14:58, 24 October 2013
Problem 21. Exercise 3.2.2 (a-d)
Problem 22. Given a set $X$ of real numbers, let $L$ be the set of all limit points of $X$. Show that $L$ is closed.
Problem 23. Show: If $X$ is both open and closed, then $X={\mathbb R}$ or $X=\emptyset$.
Problem 24. Exercise 3.2.14
Problem 25. 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: Show first that every bounded infinite set has a limit point.
