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I just read the proof of how to construct two Bernstein sets in a Polish space $X$ using the facts that there are at most $2^{\aleph_0}$ many closed subsets if $X$ is Polish and that every closed subset of a Polish space is either countable or of cardinality $2^{\aleph_0}$. Now I am asked to use the theorem of generalised recursive construction to achieve the same (this is p 176 in Just/Weese).

So I thought I could define a function $G^\ast$ such that it maps the pair $\langle f,\xi \rangle$ in $P(X \times X)$ to $(X \setminus \left \{f(\gamma)[0] \mid \gamma \in \xi \right \}) \times (X \setminus \left \{f(\gamma)[1] \mid \gamma \in \xi \right \})$. Here $f: \xi \to X \times X$ and I used $[0]$ and $[1]$ to denote the first and second coordinate of an element in the image of $f$.

Can you tell me if this is correct? Thank you for your help.

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The way I see it, there are at least a couple of problems with your function $G^*$:

  1. if $\langle x , y \rangle \in G^* ( \langle f , \xi \rangle )$, you do not ensure that $x , y \in K_\xi$;
  2. if $\langle x , y \rangle \in G^* ( \langle f , \xi \rangle )$, you do not ensure that $x \neq y$;
  3. it is not enough to ensure that if $\langle x , y \rangle \in G^* ( \langle f , \xi \rangle )$ then $x \neq f(\gamma)[0]$ for all $\gamma < \xi$ (you also need it to be different from all $f(\gamma)[1]$); similarly for $y$.

I think that if you fill in these gaps your construction will be correct.

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