Borel subset in Polish Spaces

Question

Let $f:X\rightarrow Y$ be a continuous onto mapping between Two polish spaces,then are these two statements true:-

1. Image under $f$ of the every Borel subset in $X$ is a Borel subset in $Y$.
2. Inverse image under $f$ of the every Borel subset in $Y$ is a Borel subset in $X$.

What I think is because of continuity of $f$ the second is true, but I do not have any counter-example for the second.

Answer 1

1. As Asaf mentioned, the first statement is false. An image of a Borel set under a continuous map is in not Borel (in general), and the "onto" property does not help here. A very handy isomorphism theorem - two Polish spaces are Borel isomorphic (there exists a measurable bijection with a measurable inverse) iff they are of the same cardinality. Hence, in any uncountable case you will have a Borel set whose continuous image is not Borel (in countable cases the situation is trivial). The projections are onto continuous maps, and exactly the statement whether projections of Borel sets are Borel has led to the theory of analytic sets.

2. For any topological space every continuous map is Borel-measurable: since pre-image of an open set is open (hence Borel) and open sets generate Borel $\sigma$-algebra, pre-image of any Borel set is Borel - to show measurabiltiy it is sufficient to show that pre-image of a generator is measurable. Yet again, the "onto" property is irrelevant here.