This week we learned about continuous functions, and compactness. We will have some homework about proving compactness.

1. In class, we proved that $[0,1]$ is sequentially compact, can you prove that $[0,1]^2$ in $\R^2$ is sequentially compact? (In general, if metric space $X$ and $Y$ are sequentially compact, we can show that $X \times Y$ is sequentially compact.)

2. Let $E$ be the set of points $x \in [0,1]$ whose decimal expansion consist of only $4$ and $7$ (e.g. $0.4747744$ is allowed), is $E$ countable? is $E$ compact?

3. Let $A_1, A_2, \cdots$ be subset of a metric space. If $B = \cup_i A_i$, then $\bar B \supset \cup_i \bar A_i$. Is it possible that this inclusion is an strict inclusion?

4. Last time, we showed that any open subset of $\R$ is a countable disjoint union of open intervals. Here is a claim and argument about closed set: {\em every closed subset of $\R$ is a countable union of closed intervals. Because every closed set is the complement of an open set, and adjacent open intervals sandwich a closed interval.} Can you see where the argument is wrong? Can you give an example of a closed set which is not a countable union of closed intervals? (here countable include countably infinite and finite)

5. Next week we are going to discuss open cover compactness implies sequential compactness. You can read in Pugh chapter 2 section 7.