Is it true that the closure of any set is closed? I am just assuming this fact from the word closure. My whole proof based on this fact
Proof
Let $A_1 =(a_n)_{n \in\mathbb{N}} = \{ a_n :n \geq 1\}$ and $A_2 = \{ a\}$ where $A_2$ contains all (and the only one) the limit points of $A_1$
Hence the closure of $A_1$ is $\bar{A_1} = A_1 \cup A_2$ and $\bar{A_1}$ is closed
I am thinking that I should even omit that silly last conclusion that "closure is closed"
EDIT: I just came up with a counterexample to my own argument. What if
$X = \{ (x,y) : xy < 1\}$. Technically $\bar{X} = \{ (x,y) : xy < 2\}$ is open, but it is also the closure