Proof: A space $X,\tau$ is $T_0$ iff $x\notin Cl\{y\}$ or $y\notin Cl\{x\}$ This is probably a pretty straightforward proof, but since I stand on very shaky ground with respect to all of topology and proofs in general at the moment, I would like some feedback on whether or not the following proof is adequate, and how it could be improved if it isn't.
Proposition: A space $X,\tau$ is $T_0$ if and only if given any two distinct points $x,y\in X$, either $x\notin Cl\{y\}$ or $y\notin Cl\{x\}$.
Proof:
$\rightarrow$ Suppose $X,\tau$ is $T_0$. By definition, if $x\in Cl\{y\}$, then every open set containing $x$ meets ${y}$, or in other words, every neighborhood of $x$ also contains $y$. Likewise, if $y\in Cl\{x\}$, then every neighborhood of $y$ contains $x$. But $X$ is assumed to be $T_0$, so both cannot be true. Therefore, either $x\notin Cl\{y\}$ or $y\notin Cl\{x\}$.
$\leftarrow$ Suppose $x\notin Cl\{y\}$. Then there is some neighborhood of $x$ which does not intersect ${y}$, therefore $X$ is $T_0$. 
 A: What you’ve done is basically correct, but it could be worded a bit better. In the first half of the argument, after you suppose that $\langle X,\tau\rangle$ is $T_0$, you should begin by letting $x$ and $y$ be arbitrary distinct points of $X$. Then you can use the rest of your argument for that implication without any change.
In the second half of the argument you’ve isolated the key point, but you need to say more to make the full argument clear, something like this, for instance:

Now suppose that if $x$ and $y$ are distinct points of $X$, then either $x\notin\operatorname{cl}\{y\}$ or $y\notin\operatorname{cl}\{x\}$. Without loss of generality we may assume that $x\notin\operatorname{cl}\{y\}$ (as otherwise we can simply interchange the names of $x$ and $y$). Then by definition there is an open nbhd of $x$ that does not contain $y$, and since $x$ and $y$ were arbitrary, it follows that $X$ is $T_0$.

Once you’re further along, you can skip some of the details, but when you’re just starting, it’s safest to include more detail rather than less.
