Open and Closed Set Problems using a Ball I am having trouble with these two questions. In particular, using a ball and choosing an $r$ to show that a set is open.

(a)
  $$X = \left \{  \mathbf{x} \in \mathbb{R}^d | \:  ||\mathbf{x}|| \leq 1 \right \} .$$

So, $X$ is closed if its compliment $X^c$ is open. So if I can show that $X^c$ is open, then it follows that $X$ is closed. I'll start with the definition I am using for a ball.

The ball about $\mathbf{a}$ in $\mathbb{R}^n$ of radius $r$ is the set
  $$B_{r}(\mathbf{a})= \left \{ \mathbf{x} \in \mathbb{R}^n : || \mathbf{x} - \mathbf{a} || < r\right \}$$
  A subset $U$ of $\mathbb{R}^n$ is open if for every $\mathbf{a} \in U$, there is some $r=r(a) > 0$ such that the ball is contained in $U$. 

So, I have a set $X^{c} = \left \{  \mathbf{v} \in \mathbb{R}^d | \:  ||\mathbf{v}|| > 1 \right \} $. It consists of points whose lengths are longer than 1. Let $\mathbf{a} \in X^{c}$, then $|| \mathbf{a} || > 1$. Now, using $B_{r}(\mathbf{a})$ as it is written above, how do I find an explicit formula for $r>0$? Intuitively it seems to make sense if $\mathbf{x}$ is in $X$, then $0<|| \mathbf{x} - \mathbf{a} ||$ as $||\mathbf{x}|| \neq  ||\mathbf{a}||$. I am not sure how to proceed.

(b)
  $$X = \mathbb{R}^2 \setminus \left \{  \mathbf{x} \in \mathbb{R}^2 | \mathbf{x}=(x,0) \: \right \}$$

I know that this set is open, but it again comes down to choosing some $r$ and using a ball. The set is $\mathbb{R}^2$ less a line across the x-axis - that is, any $x$ and any $y$ with $y \neq 0$. The line has no height, so it should be easy to show that for any $\mathbf{a} \in X$  there exists $r>0$ such that $B_r(a) \subseteq X$.
I've done lots of scratch work and diagrams etc. but I just can't seem to put this concept together. Any help and clarification would be appreciated.

For (a) draw a plane figure. Choose a point $a\notin X$. How large a circle can you draw around $a$ such that its interior does not intersect $X\>$? Use some form of the triangle inequality to show that what you see is indeed true.
With (b) it's the same thing. Choose a point $a:=(a_1,a_2)\notin X$. How large a circle can you draw around $a$ such that its interior does not intersect $X\>$?
