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Math Help - Sets of Reals

  1. #1
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    Lightbulb Sets of Reals

    Prove that there exists a X \in \mathcal{P} (\omega) with \left| X \right|= 2^{\aleph_0} such that every member of X is infinite, but a \cap b is finite for all distinct a,b \in X.

    Prove that for every X \subset \mathbb{R} there are disjoint subsets Y,Z such that X = Y \cup Z, Y is countable, and for no pair a,b \in \mathbb{R} is the set \left\{ c \in Z : a < c < b \right\} nonempty and finite.
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    Quote Originally Posted by syme.gabriel View Post
    Prove that there exists a X \in \mathcal{P} (\omega) [I'll assume you mean \color{red}X \subset \mathcal{P} (\omega).] with \left| X \right|= 2^{\aleph_0} such that every member of X is infinite, but a \cap b is finite for all distinct a,b \in X.

    Prove that for every X \subset \mathbb{R} there are disjoint subsets Y,Z such that X = Y \cup Z, Y is countable, and for no pair a,b \in \mathbb{R} is the set \left\{ c \in Z : a < c < b \right\} nonempty and finite.
    For the first one, let f:\mathbb{Q}\rightarrow\omega be an enumeration of the rationals. For every irrational number x, let (a^{(x)}_n) be a sequence of rationals converging to x, and let S_x = \{f(a^{(x)}_n):n\geqslant1\}. Then X will be the collection of all these sets S_x.

    For the second one, don't you just take Y to be the set of isolated points in X, or am I missing something?
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