# Measurably-isomorphic subsets of polish spaces and the continuum hypothesis

In Theorem 2.7 in the following notes, we seem to assume the following statement.

Let $(\Omega,\mathcal F)$ be a Polish space, and $A\in\mathcal F$ an uncountable set. Then there exists a bijection $\phi:(A,\mathcal F |_A)\to([0,1],\mathcal B([0,1]))$ such that $\phi$ and $\phi^{-1}$ are both measurable.

Now, in the book Probability Measures on Metric Spaces by Parthasarathy et al, Theorem I.2.12 states the following:

Let $X_1,X_2$ be separable complete metric spaces, and $E_i\subset X_i$ Borel subsets, for $i=1,2$. Then $E_1$ and $E_2$ are isomorphic (in the sense described above) iff they have the same cardinality.

But do we know that any uncountable Borel-measurable $A\subset\Omega$ where $A$ is polish necessarily has cardinality at least that of the continuum?

• This is Theorem 2.8 in the book by Parthasarathy (not et al). – Michael Greinecker Apr 9 '18 at 8:34

Every Borel subset (and in fact every analytic subset) of a Polish space either is countable or has a perfect subset. In particular, an uncountable Borel subset in a Polish space has the cardinality of the continuum. This can be found, for example, in Moschovakis's book "Descriptive Set Theory" as Corollary 2C.3. The result apparently goes back to Suslin and Lusin in 1917.

• Moschovakis's book was the first place I looked, and I didn't look further. I expect you can also find the result in other standard texts, like Jech's "Set Theory" and Kechris's "Classical Descriptive Set Theory". – Andreas Blass Apr 8 '18 at 23:46
• It's Theorem 13.6 in Kechris. – Nate Eldredge Apr 9 '18 at 5:44