Then, $\displaystyle d_\infty(\bold {a},\bold {x}_{0})< \varepsilon $.

So, $\displaystyle max \{ |a_1 - x_1|, |a_2 - x_2| \} <\varepsilon $.

$\displaystyle \Rightarrow |a_1 - x_1| < \varepsilon \ \ or \ \ |a_2 - x_2|< \varepsilon $

$\displaystyle \Rightarrow -\varepsilon + x_1 < a_1 < \varepsilon + x_1 \ \ or \ \ -\varepsilon + x_2 < a_2 < \varepsilon + x_2 $

Thus, if we add those inequalities, we have $\displaystyle -2\varepsilon +x_1 + x_2 <a_1+a_2 \ \ or \ \ a_1+a_2 < 2\varepsilon +x_1 + x_2$

Then, choose $\displaystyle \varepsilon = min \{\frac{1}{2}(x_1 + x_2),-\frac{1}{2}(x_1 + x_2)\}$ for $\displaystyle a_1 + a_2 < 0 \ \ or \ \ a_1 + a_2 > 0$.

Thus, $\displaystyle \bold {a} \in \Omega$ and $\displaystyle d_\infty(\bold {a},\bold {x}_{0}) \subset \Omega$

Hence, $\displaystyle \Omega$ must be open.

**Comment**
Is the above working correct?

Is using the infinity distance function $\displaystyle d_\infty(\bold {a},\bold {x}_{0})$ instead of the euclidean distance function, mathematically correct? Could I have used something like $\displaystyle d_1(\bold {a},\bold {x}_{0})$ instead?

Is there anyway to simplify the working out? What is the best way choosing an epsilon?

Thanks in advance.