[Math] Closure proof and closure properties.

proof-writingreal-analysis

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

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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

Best Answer

Well, since (the or one of the equivalent) definition of closure of a set $\,A\subset X\,$ , in a topological space $\,X\,$ is

$$\overline A:=\bigcap_{A\subset B\,,\,B\,\text{closed}}B$$

and any intersection of closed sets is closed, then $\,\overline A\,$ is closed, too.

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