Proof on if a set is discrete

Proof on if a set is discrete

I would like to know how well I answered the following proof: was it
concise? Was it elaborate/rigorous? Did I use incorrect notation?
Also, I would like to know how to describe a metric space with this
notation
(http://upload.wikimedia.org/math/1/b/f/1bfabe2211af146e4d250eeb9f921080.png)instead
of my messy one. I don't know how to do that with $LaTeX$; an edit would
be greatly appreciated.
I would also like to know if the set is a $T_{1}$ space, such that every
pair of distinct points contains an open ball not containing any other
point. By definition, I suppose it is such a space, but I would like
verification.
Onward,
Prove the following metric space $M$ is discrete: $$D(x, y) : (1 if x
\not= y), (0 if x = y)$$
The metric is discrete if every subset is open. This needs proof.
Suppose there exists a subset $x, y \in M$. Define $$x \in E, y \in F : E
\cap F = \emptyset : E, F \subset M$$. Also, there are distinct points
such that $x \not= y$, otherwise the proof is trivial.
Since $y \not\in E$ and $E$ is not closed, ${x}$ is an isolated point of $E$.
Thus there exists a neighborhood $B_{r}(x) : y \not\in B_{r}(x) : x \not=
y$. QED.