To assess my basic understanding of metric spaces, I am doing the following exercise. However, I don't know if it is correct.
Let $X=\mathbb{R}^2$ and let $d(x,y)$ the Hamming distance, i.e. $d(x,y)$ is number of places where $x$ and $y$ have different entries.
a) Sketch $\mathcal{B}_{0.5}((0,0))$ and $\mathcal{B}_{1.5}((0,0))$;
b) Does the sequence $x_n=(1/n,0)$ converges to zero?
c) let $A:=\{(x,y): 0\le x \le 1, 0\le y \le 1\}$. Find $A^o$;
d) Find all the dense set in $X$.
My Solution
a) Given that we are in $\mathbb{R}^2$, then $d(x,y)=\{0,1,2\}$, i.e. two vector can have at most two different entries. In order to design the ball, I have to find the set of points for which $d((0,0),(x,y))$ is less than 0.5 and less than 1.5. Here are the two balls that I have found Ball of radius 0.5 Ball of radius 1.5
b) $lim_{n\to\infty} d((1/n,0),(0,0)) = 0$. The sequence converges.
c) $A^o = \{(x,y): 0< x < 1, 0< y < 1\}$. Infact, any ball centered in $(0,0)$ contains some points with coordinate $x<0$ and $y<0$ (thus they belong to the ball but not to the set $A$). A ball in $(0,1)$ contains some point $y>1$ and a ball in $(1,0)$ contains some point $x>1$.
d) Every closed subset of $\mathbb{R}^2$ is dense.