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 that every non-empty open set is the countable union of pairwise disjoint open intervals: 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) for all $m,n\in\mathbb{N}$: $I_m\cap I_n=\emptyset$ or $I_m=I_n$.