Metric space and topology. $\gdef\In{\subset}$

1. Consider the metric space $(X = \Z, d_X(x,y) = |x-y|)$. Write down the open ball $B_3 (2)$, $B_{1/2}(2)$. Is the subset $\{2\}$ open in $X$? Is it closed in $X$? Explain.

2. Consider the metric space $(X = [0,1) \In \R, d_X(x,y) = |x-y|)$. Write down the open ball $B_{3} (0)$, $B_{1/3}(0)$. Is the subset $[0,1)$ open in $X$? closed in $X$? Is the subset $[1/3, 1)$ open in $X$? closed in $X$?

3. Construct a subset $E \In [0,1]$, such that the limit points of $E$ is $E' = \{0, 1\}$. Optional: Is it possible to construct $E$, such that $E' = \{0, 1, 1/2, 1/3, \cdots \}$?

4. Let $X =\R$, let $d(x,y) = \sqrt{|x-y|}$. Is $d(x,y)$ a distance function on $X$?

5. Let $X$ be a metric space, and $E \In X$ be any subset. Prove that $\overline{E^c} = (E^o)^c$, where $E^o$ means the set of interior points in $E$.

Optional. Let $(X, d)$ be a metric space, $k \geq 1$ be an integer. Let $Conf_k(X) = \{ S \In X, |S|=k\}$, i.e., an element in $Conf_k(X)$ is subset $S \In X$ consisting of $k$ points. For example, let $X = \R$, $k=2$, then $S = \{ 2, -1.1 \}$ is an element in $Conf_2(X)$. Can you put a metric on $Conf_k(X)$ using $d$?