This question stems from Problem 10, section 4.2 and Problem 9, section 13.1 of Real Analysis and Probability written by Dudley. Similar question has been posed in: Problem 10, section 4.2 of R.M. Dudley, Real Analysis and Probability.
In Problem 10, section 4.2, we are asked to prove: Let $f$ be a Borel measurable function from a separable metric space $X$ onto a metric space $S$ with metric $e$. Show that $(S, e)$ is separable.
And in Problem 9, section 13.1, we are asked to prove: Let $(S, d)$ be a separable metric space and $(T, e)$ a metric space. Let $f$ be a Borel measurable function from $S$ into $T$ . Assuming the continuum hypothesis, prove that the range $f[S]$ is separable.
I'm very confused about the difference between these two statements. It seems that the proof of the previous one does not need the continuum hypothesis. But I can't find such a proof. In my point of view, we can't prove Problem 10 in section 4.2 without assuming continuum hypothesis, if we follow the hint given by Dudley: If $S$ is not separable, then show that for some $ε>0$, there is an uncountable subset $T$ of $S$ with $d(y, z)>ε$ for all $y\neq z$ in $T$. Use Problem 9 (the statement and proof of Problem 9 is given in: Problem 9, section 4.2 of R.M. Dudley, Real Analysis and Probability) to get a measurable function $g$ from $X$ onto $T$. All $g^{-1}(A) (A \subset T)$ are Borel sets in $X$.
Thanks in advance to your help!
