Let $(E, <)$ be a poset. We can endow $E$ with the order topology, and thus view it as a measurable space $(E,\mathcal{E})$ with the Borel $\sigma$-algebra.
Is the set $M$ of maximal elements of $E$, $M:= \{x : y \leq x \text{ for all } y \}$ measurable with respect to $(E,\mathcal{E})$?
It is quite straightforward to write $M = \bigcap_{y \in E} (-\infty,y)^c$, where $(-\infty,y) := \{ x \in E: x < y\}$. Hence, one sufficient condition is that $E$ is countable. However, I am wondering if there are any more necessary or sufficient conditions we can impose.
Countability is not necessary. For example, $M$ is measurable for $[0,\infty]$ with the usual order, since $M = \{\infty\} = \bigcap_{n \in \mathbb{N}} [0,n]^c$. More generally, $M$ will be countable if we can determine whether an element is maximal by only 'comparing' it against countably many other elements, but I am not sure if this is a notion that has been formalised.