cI have this topology on $\mathbb{R}^2$ $$\sigma=\big\{\emptyset,\{(3,0)\}\big\}\cup\big\{\bigcup_{a\in A}\Omega_a : A\subseteq \mathbb{R}\cup\{\infty\}\big\}$$
where $$\Omega_a=\{(x,y)\in\mathbb{R}^2 : y-ax+3a=0\},\Omega_{\infty}=\{(x,y)\in \mathbb{R}^2 : x=3\}$$
I need to find the adherence of the $C=\{(x,y)\in \mathbb{R}^2 : x^2+y^2=4\}$
I see that
$(3,0)\notin \rm cl\{C\}$
the two lines $\Omega_{\dfrac{2}{\sqrt{5}}}$ and $\Omega_{\dfrac{-2}{\sqrt{5}}}$ are tangent to C, so $$\forall (x,y)\in \big[ \bigcup_{\dfrac{-2}{\sqrt{5}}\leq a\leq \dfrac{2}{\sqrt{5}}}\Omega_a\big]\setminus\{(3,0)\}, (x,y)\in \rm cl\{C\}$$
$C\subset \rm cl\{C\}$
Is it true if i say that $$\rm cl\{C\}=C\cup \big[ \bigcup_{\dfrac{-2}{\sqrt{5}}\leq a\leq \dfrac{2}{\sqrt{5}}}\Omega_a\big]\setminus\{(3,0)\}$$