I'm confused by a subtle point in the definition of analytic sets. Suppose I have a Polish space $X$. Now I start with the collection of Borel sets in $X$ and take all their continuous images in $X$. Do I get the entire family of analytic sets in this way? In other words, can I say in good conscience that the analytic sets in $X$ are the continuous images of the Borel sets in $X$?
Let me state the question another way. By definition a set $A\subseteq X$ is analytic if it is the image $A=f(B)$ of some Borel set $B\subset P$ in some Polish space $P$ using some continuous mapping $f:P\to X$. I don't like referring to an external space $P$. What happens if I try to simplify the definition by requiring $P=X$; will I still get all the analytic sets?