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Is there a "standard" notation to denote the set of all finite subsets of $\mathbb{N}$? (or any set, not just $\mathbb{N}$)

Thanks

  • 0
    How about "Give X the discrete topology and consider $C_c (X,\{0,1\})$? :D2011-08-04

3 Answers 3

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Several possible notations for $\{A\subseteq\omega\mid |A|<\omega\}$:

  1. $[\omega]^{<\omega}$
  2. $P_\omega(\omega)$
  3. $\operatorname{Fin}(\omega)$

Where, of course, $\omega=\mathbb N$.

And as usual my advice on the matter: When in doubt, open with "We denote by [the chosen notation here] the set ..."

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    @Qiaochu: I agree that it is not a very welcoming notation. It is a *very* common notation in set theory. For example, $\kappa$ is $\lambda$-supercompact if there exists a fine and normal $\kappa$-complete measure on $\mathcal P_\kappa(\lambda)$.2011-08-04
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You can find various notations, as mentioned in coments. (I doubt there is some generally accepted notation.)

  • You can find $[\omega]^{<\omega}$, e.g. here, which can be considered as a special case of $[A]^{<\kappa}$ - which denotes all subsets of $A$ of cardinality less then $\kappa$, see e.g. p.18 of the same book. In your case you could use $[\mathbb N]^{<\omega}$.

  • You can find $\mathrm{Fin}$, e.g. here and here

  • You can find $\mathbb N^{[<\infty]}$, e.g. here.

  • Hindman and Strauss use $\mathcal P_f(\mathbb N)$ in this book, which is similar to Qiaochu's suggestion $\mathcal P_{\mathrm{fin}}(\mathbb N)$.