Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 3 / 21F / b / Solution

Writewith each $U_n$ open. The closed sets $A$ and $X\setminus U_n$ are disjoint. By the Urysohn lemma, choose a continuous $f_n:X\to [0,1]$ withThe uniformly convergent seriesdefines a continuous map to $[0,1]$. It vanishes on $A$. If $x\notin A$, then $x\notin U_n$ for some $n$, so $f_n(x)=1$ and $f(x)>0$. Thuswhich is the closed G-delta set as a zero set construction.