Defining closed and open sets with quantifiers. I'm taking an intro to analysis course, and, given the following definitions of closed and open sets, I was wondering whether my attempt at quantifying them is correct.

Definition A set $A\subset R$ is closed iff every accumulation point of $A$ belongs to $A$  $$\overset{{?}}{\iff}\forall{x}\left(\forall\varepsilon&gt;0\exists{a\in{N(x,\varepsilon)\cap (A\setminus{\{x\})}}}\implies x\in{A}\right)$$
A set $E\subset R$ is open iff for every $x\in A$ there is a neighborhood  $N(x,\varepsilon)$ of $x$ such that $N\subset A$ $$\overset{{?}}{\iff}\forall{x}\in{A}\exists\varepsilon&gt;0:N(x,\varepsilon)\subset{A}$$
Let $A\subset \mathbb R $.

$A $ is colsed in $\color {red}{\mathbb R} \iff $

$$\overline {A}\subset A \iff $$

$$(\forall x\in \color {red}{\mathbb R}) $$

$$\Bigl ( \forall \epsilon&gt;0 \; (x-\epsilon,x+\epsilon)\cap A \neq \emptyset \implies x\in A\Bigr) $$
Let $A\subset \mathbb R $.

$A $ is colsed in $\color {red}{\mathbb R} \iff $

$$\overline {A}\subset A \iff $$

$$(\forall x\in \color {red}{\mathbb R}) $$

$$\Bigl ( \forall \epsilon&gt;0 \; (x-\epsilon,x+\epsilon)\cap A \neq \emptyset \implies x\in A\Bigr) $$