View source for CRN 12109: HW 5


'''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.