If $(X,d)$ is a metric space, why are $X$ and $\emptyset$ open sets? My definition of an open set $A$ in a metric space $X$ is that $\forall x\in A\exists y>0: B(x,y) \subseteq X$. How does this theoretically coincide with $X$ and $\emptyset$ being open, looking formally at this?
 A: For $\emptyset$: 
$$(\forall x \in \emptyset)(\exists r>0) \: B_r(x) \subseteq \emptyset$$ 
is vacuously true, because there are no points in the empty set. So $\emptyset$ is open.
For $X$, there is a catch that you have to keep in mind: $B_r(x)$ is defined to be $\{ y \in X : d(x,y)<r \}$. That is, by definition every member of $B_r(x)$ is contained in $X$. Thus $B_r(x) \subseteq X$ comes for free, so $X$ is open. 
This last point can be a little bit tricky when you deal with familiar examples, like subsets of $\mathbb{R}$. For instance this definition implies that $(1/2,1]$ is an open subset of $[0,1]$. This is correct even though $(1/2,1]$ is not an open subset of $\mathbb{R}$ itself.
A: If $x\in X$ then certainly $B(x,1)\subseteq X$, i.e. all points in $X$ that are less than $1$ unit away from $x$ are in $X$.  To express it in the notation you used in the question, one can take $y$ to be $1$.  Actually any positive number in place of $1$ suffices.
As for the empty set, it has no members; thus "all of its members are purple" is true (as long as "purple" means something).  Are all cell phones in the classroom turned off?  If there are no cell phones in the classroom then the answer is "yes". "All cell phones are turned off" means there are none that are not turned off.  "All points in the empty set are withing $1$ unit away from whichever point" means there are no points in the empty set that are not within that distance.  And that is true.  Hence the empty set is open.
A: $\emptyset$ is open: You have $\forall x \in \emptyset \ldots$ which is true because each forall statement over an empty set is by convention true:
$X$ is open: Not that for each $x\in A$ and $y > 0$ you have $B(x,y)\subseteq X$ because $B(x,y)$ is by definition a subset of $X$ ($B(x,y)=\{z\in X: d(x,z) < y\}$).
