Notation for infinite coinfinite subsets.

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.

• My understanding is that a subset $S$ of a set is cofinite if $S^c$ is finite, and similarly, $S$ is coinfinite if $S^c$ is infinite. I guess this is not the definition you have in mind. Could you please give your definition of these notions? – J.-E. Pin Oct 17 '19 at 10:11
• Do you mean "infinite coinfinite"? These are the sets that split $\omega$. – Jonathan Oct 17 '19 at 12:05
• @Jonathan Oops, indeed, I mean infinite coinfinite – Vsotvep Oct 17 '19 at 15:13
• @J.-E.Pin I meant to write infinite coinfinite, sorry for the confusion. – Vsotvep Oct 17 '19 at 15:15
• I don't think there is a standard notation for this, annoyingly. I've seen things like "$\mathcal{P}_{biinf}(X)$" or "$[X]^{biinf}$" or similar, but not something established. – Noah Schweber Oct 17 '19 at 15:25

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 $$[X]^{\geq \aleph_0} - \text{Fr}(X)$$.