Can I have feedback on my proof, please?
Prove the Michael line topology, $T_\mathbb{M}=\{U \cup F: U$ is open in $\mathbb{R}$ and $F\subset \mathbb{R}\setminus \mathbb{Q}\}$ is $T_{2}$ (Hausdorff).
Let $a,b \in \mathbb{R}$. WLOG, let $a<b$. Let $x \in (a,b)$. Then $a<x<b$. Clearly, $(-\infty , x), (x, \infty)$ are open disjoint subsets of $T_\mathbb{M}$. Furthermore, $a \in (-\infty , x)$ and $b \in (x, \infty)$, thus, $T_\mathbb{M}$ is $T_{2}$ or Hausdorff.
Your proof is correct: you separate the points with sets that are standard-open (in the reals) and because the Michael line topology has as its topology a superset of the usual topology (because we can always take $F = \emptyset$), these standard-open sets are still open in the new topology and are as required.