$\gdef\In{\subset}$

1. Determine whether following subset $S$ of metric space $X$ is (a) open or not (b) closed or not © bounded or not (d) compact or not. (You may use Heine-Borel theorem for $\R^k$)

• $X = \R$ with usual metric. $S = \Q \cap [0,1]$.
• $X = \R^2$ with Euclidean metric. $S = \{ \vec x \in \R^2 \mid |\vec x| = 1 \}$

2. True or False, give your reasoning or give an counter-example.

• Let $(X, d)$ be a metric space, then any finite subset $S \In X$ is closed.
• Let $X = \Q$ with the usual metric. Then, any closed bounded subset $S \In X$ is compact.

3. (Open and closed subset are relative notion) Let $(X, d)$ be a metric space. $U \In Y \In X$ any subset. Prove that

• If $Y$ is open relative to $X$, then $U$ is open $Y$ if and only if $U$ is open with respect to $X$.
• (Optional) If $Y$ is closed relative to $X$, then $U$ is closed relative to $Y$ if and only if $U$ is closed relative to $X$.

4. (Compactness is absolute notion) Determine whether the following metric space is compact.

• $X = [0,1] \cap \Q$, with the induced metric from $\R$.
• (Optional) $X = \cup_{n=1}^\infty \{ (x_1,x_2) \in \R^2 \mid (x_1-1/n)^2 + x_2^2 = (1/n)^2 \}$, with induced metric from $\R^2$.

Hint: In class we proved the following result: if $(X, d)$ is a metric space, $K \In Y \In X$, then $K$ is a compact subset relative to $Y$ if and only if $K$ is a compact subset relative to $X$.

5. Give examples.

• An (infinite) countable subset in $\R$ that is compact.
• An (infinite) countable subset $S \In \R$, such that $S \In S'$, ie, every point of $S$ is a limit point of $S$.