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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.

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  • $\begingroup$ 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? $\endgroup$ – J.-E. Pin Oct 17 '19 at 10:11
  • $\begingroup$ Do you mean "infinite coinfinite"? These are the sets that split $\omega$. $\endgroup$ – Jonathan Oct 17 '19 at 12:05
  • $\begingroup$ @Jonathan Oops, indeed, I mean infinite coinfinite $\endgroup$ – Vsotvep Oct 17 '19 at 15:13
  • $\begingroup$ @J.-E.Pin I meant to write infinite coinfinite, sorry for the confusion. $\endgroup$ – Vsotvep Oct 17 '19 at 15:15
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    $\begingroup$ 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. $\endgroup$ – Noah Schweber Oct 17 '19 at 15:25
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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 $[X]^{\geq \aleph_0} - \text{Fr}(X)$.

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