I have this topology on $\mathbb{R}^2$ :$$\{\emptyset\}\cup\{\Omega_n,~n\geq 0\}$$
where $$\Omega_n=\{(x,y)\in\mathbb{R}^2, |x|\geq n\}$$
I want to find the set of accumulation points of the point $(2,-2)$ denoted by $\{(2,-2)\}'$
first I found the closure with is $\mathbb{R}^2\setminus\Omega_3$
Then I found that $\{(2,-2)\}'=[\mathbb{R}^2\setminus\Omega_3] \setminus\{(2,-2)\}$
I'm right?
Thank you
The closure is correct. It’s the smallest closed set that contains $(-2,2)$. As a consequence the set of limit points is also correct, because in any space $\{x\}’=\overline{\{x\}}\setminus \{x\}$, as $x$ cannot be a limit point of $\{x\}$ by definition, and all limit points lie in the closure.