If $$\mathcal{F}=\{(r_1,r_2)\in\mathbb{R}^2 : r_1^2+r_2^2<1,(r_1+1)^2+r_2^2\geq 1,-\frac{1}{2}\leq r_2<\frac{1}{2}\},$$ $e_1=(1,0)$ and $Y=\{(r_1,r_2)\in\mathbb{R}^2 : r_1\leq 0\}$ I need to determine if $$0\in\textrm{int}_{-}(\overline{\mathcal{F}}-e_1)$$ where $\textrm{int}_{-}A$ means interior of a set $A$ in a relative topology on $Y$.
What is the answer to this and how can it be proved. Thanks in advance.
To deal with such questions is very helpful to draw the considered sets on a picture. When I drew it for your question, I saw that $0\not\in\textrm{int}_{-}\left(\overline{\mathcal{F}}-e_1\right)$, because a sequence $\left\{\left(0,\tfrac 1n\right):n\in\Bbb N\right\}$ of $Y\setminus \left(\overline{\mathcal{F}}-e_1\right)$ converges to $0$.