Closure and interior of sets in this topology?

Question

i have a topology $\theta$ on $\mathbb{R}^2$ defined with it's basis $\mathcal{B}$ is the family of all sets $$D_{a,b,c}=\{(x,y)\in\mathbb{R}^2, y>ax+b; y>-ax+c\}$$ where $a>0$ and $b,c\in \mathbb{R}$

I want to find the closur and the interior of the following sets \begin{align} & A=\{(x,y)\in\mathbb{R}^2, y^2+x^2<1\},\\ & B=\{(x,y)\in\mathbb{R}^2, x>0\},\\ & C=\{(0,y)\in\mathbb{R}^2, y>0\}. \end{align}

It is clear for me that $\overset{\circ}{A}=\emptyset=\overset{\circ}{B}$ and $\overset{\circ}{C}=C$

But i have a problem with the closure of each set?

Thank you

Answer 1

It seems to me that the closure of $A$ will look like this and that the closures of $B$ and $C$ will be the entire plane.