Call a subset $A\subset X$ of an infinite set $X$ coinfinite if $X\setminus A$ is infinite.
Is there a standard way to denote the infinite coinfinite subsets of an infinite set (in particular of $\omega$)?
I have made up $\mathrm{Spl}(\omega)$, motivated by it being the set of those subsets of $\omega$ that split $\omega$, but I was wondering if there is a more common notation.
A standard notation for the class of infinite subsets of $X$ is $[X]^{\geq \aleph_0}$.
On the other hand, the Fréchet filter (class of cofinite subsets) of $X$ can be denoted by $\text{Fr}(X)$.
Therefore, your set can be denoted by $[X]^{\geq \aleph_0} - \text{Fr}(X)$.