Let $X=\mathbb{R}$ and $S$ be the set of irrational numbers. Find the closure of $S$ in the following topologies for $\mathbb{R}$ Let $X=\mathbb{R}$ and $S$ be the set of irrational numbers. Find the closure of $S$ in the following topologies for $\mathbb{R}$ (a) The standard topology of $\mathbb{R}$ (b) The discrete topology of $\mathbb{R}$ (c) The indiscreet topology of $\mathbb{R}$ (d) The topology of the accounting complement My intent: (a)$\mathbb{R}$, (b) $\mathbb{I}$, (c)$\mathbb{R}$, (d) $\mathbb{Q}$ This is OK? Thank you very much.

As I stated in the comments, your answers to (a), (b), and (c) are perfect.

For (d), we need to find the closure of the irrational numbers $S$ in the cocountable topology. That is, we must find the smallest closed set containing the irrational numbers. A set is closed in the cocountable topology iff it is countable or is the whole space. Since $S$ is uncountable, it cannot be contained by any countable set. Therefore its closure is the whole space $\mathbb{R}$.