# 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). Apr 9 '18 at 8:34