Let $X$ be the set of real numbers.
- Let a topology be $T_r$ generated by $\{(a,\infty): a$ is a real number$\}$ I have to find the derived set of $(0,1)$ in this topology. My answer is $X$, as any neighborhood of any number has an intersection with $(0,1)$ other than the number. Also the topological space $(X, T_r)$ is not Hausdorff, I think it is not $T_1$ and also not $T_2$. Am I right?
- Let another topology on $X$ be $T_l$ , the lower limit topology. I have to find the interior of $(0,1]$. My answer is $(0,1)$. Are my answers right? If not hope to get corrected. Thank you.