Let $\mathbb{R}_{\tau}$ be the set of real numbers with topology $\tau = \{(-x,x)| x>0\} \cup \{\emptyset, \mathbb{R}\}$ and $\mathbb{R}_{\tau} \times \mathbb{R}_{\tau}$ be the product topology on $\mathbb{R}^2$.
a) Prove that $A = \{(x,y) \in \mathbb{R}^2 | x^2 + y^2 < 1\}$ is open in $\mathbb{R}_{\tau} \times \mathbb{R}_{\tau}$.
b) Find $\overline{A}$. Justify your answer.
c) What functions $f: {\mathbb{R}_{\tau}}^2 \rightarrow \mathbb{R}$ are continuous? Here $\mathbb{R}$ has the standard topology and ${\mathbb{R}_{\tau}}^2 = \mathbb{R}_{\tau} \times \mathbb{R}_{\tau}$ has the product topology.
Please help!