Proposition from the book "Convex analysis and measurable multifunction "

Question

Please , what is $F_{\sigma}$? And all $U$ can be writen as $\cup F_n$, or only open set ?

Thank you

Answer 1

An $F_\sigma$ set is a set which is the countable union of closed sets. Every open set in a metric space can be written as such, but not every set is $F_\sigma$.

To see that the sets $F_n$ if the proof are closed, consider their compliment:

$$ F_n^c=\{x\in X\vert d(x,X-U)<\frac{1}{n}\} $$ These sets are open, since if $y\in F_n^c$, $d(y,X-U)<1/n$ and the same inequality will hold for all $x$ in a sufficiently small neighborhood about $y$.