View source for CRN 11982: HW 6

'''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 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 $m\neq n$.