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Let $X$ be a set such that $|X|=\aleph _0$. I need to find a family of sets $\mathcal{F}$, of subsets of $X$ such that $|\mathcal{F}|=|\mathbb{R}|$.

I saw a couple of examples of Specific X but I can't construct a family of sets which hold the condition for a general set X with $|X|=\aleph _0$.

edit: I forgot to mention an important condition that $\mathcal{F}$ must hold: for every $A,B\in \mathcal{F}$ the intersection $A\cap B$ is finite.

Vegetal605
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  • Do you know what are the cardinalities of $\Bbb R$ and $\mathcal P(X)$? Or is there some additional requirement on $\cal F$ that you forgot to mention? – Asaf Karagila Sep 02 '15 at 08:48
  • @Asaf Karagila In my course we denote the cardinality of $\mathbb{R}$ by $\aleph$ and if $|X|=\aleph _0$ then I know that $|\mathcal{P}(X)|=|\mathbb{R}|=\aleph$, and the only condintion on $\mathcal{F}$ is that it is a family of sets of subsets of $X$ – Vegetal605 Sep 02 '15 at 08:50
  • Is $\mathcal P(X)$ not a sufficient answer, then? – Asaf Karagila Sep 02 '15 at 08:58
  • @AsafKaragila only now I realized I forgot to mention the extra important condition that $\mathcal{F}$ must hold. I edited the question. sorry :( – Vegetal605 Sep 02 '15 at 09:10

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