As title says. I’m trying to learn some descriptive set theory but I don’t quite see this.
I want to use the following:
Given $X, Y$ Polish spaces, $f:X\to Y$ continuous, if $f(X)$ is uncountable there is a subset $K\subseteq X$ homeomorphic to Cantor space on which $f$ is injective.
I can reduce to the case where $f(U)$ uncountable for $U$ open in $X$, and Kechris says to show $\{K\in K(X) : f \text{ injective on K}\}$ is a dense $G_\delta$ set, in the Vietoris topology on $K(X)$ the compact subsets of X.
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How do I show this? I suspect “Lusin schemes” might be useful but I don’t really understand this technology. Other approaches are also welcome.
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Why does this give the result? Being $G_\delta$, this set is then Polish (right?), but why does this yield an uncountable K (which I understand would be sufficient)
Thank you
Best Answer
For 2. $K(X)$ is Polish, the intersection of countably many dense open sets is dense and hence nonempty, so there are plenty of $K$ on which $f$ is injective.
For 1. I would try something like this: take a countable base, $\mathcal{B}$, for the topology of $X$. For a pair $\langle B, C\rangle$ of elements of $\mathcal{B}$ with disjoint closures let $$U(B,C)=\{K \in K(X): f[K\cap B]\cap f[K\cap C]=\emptyset\}\text{.}$$
Then prove that $U(B,C)$ is open and dense.