CRN 12109: HW 5
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| − | '''Problem 25.''' A set $X$ is called '' | + | '''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 | + | 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. | ||
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.
