1
$\begingroup$

I'm trying to understand a proposition of A. Kechris in chapter 8 of his Classical Descriptive Set Theory, in which given a non empty metrizable separable space $X$ that is dense in a Polish space $Y$, we have to prove that if $X$ is comeager in $Y$ then $X$ is Choquet i.e. the second player has a winning strategy in the Choquet game on $X$. Of course, $X$ contains an intersection of open and dense subsets $W_n$ of $Y$ , that also must be dense because $Y$ is Baire. What is the winning strategy for II? I put $V_n=U_n\cap W_n$ but I'm not sure that is correct.

$\endgroup$
  • 1
    $\begingroup$ Perhaps an indication of how the Choquet game goes might be of use (that is, where $U$s and $V$s come from). $\endgroup$ – Pedro Sánchez Terraf Mar 5 '16 at 21:57
0
$\begingroup$

Just a reminder: The Choquet game on $X$ has two players ($I$ and $II$) who alternatively choose open sets in $X$ (resp., $U_n$ and $V_n$ with $n\in\mathbb{N_0}$) subject to the condition $U_n\supseteq V_n\supseteq U_{n+1}$ for all $n\in\mathbb{N_0}$. Player $II$ wins a round if $$ \bigcap_n U_n =\bigcap_n V_n \neq \emptyset. $$

In this context, your proposed strategy won't work as it stands, but almost does. Assume $X= Y = (0,1) =W_n$ for all $n$ and $I$ plays $U_n=(0,\frac{1}{n})$. This example makes it clear that you should use the completeness of $\bigcap_n W_n$ somewhere.

Getting into the argument, $G = \bigcap_n W_n \subseteq X$ is Polish with the relative topology; choose a complete metric $d$ for it. Now the strategy for $II$ is to take $V_n\subseteq U_n\cap W_n$ such that $\overline{V_n}\subseteq U_n$ and $\mathrm{diam}(V_n)<2^{-n}$. Since $W_n$ is open and dense, we can ensure that each such $V_n$ will be nonempty; let $x_n\in V_n$. It is easy to see that $\{x_n\}_n$ is Cauchy in $G$, so it converges to $x\in G$. This $x$ must belong to $$ \bigcap_n \overline{V_n} =\bigcap_n U_n, $$ hence II wins this play.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.