$\bf Ques1:$ In 1st line of [$\bf \text{ Solution of Problem 19(i)}$] if $x\notin X_n$ then there may be $n-1$ members of the family $\gamma$ containing $x$. In that case how we construct a open set, containing $x$ disjoint from $X_n$?
Observation:(1) $X_1$ is closed, $X_1\cup X_2\cup ...\cup X_n$ is closed and all $X_n$ are disjoint.
(2) Each $F_n(x)$ is clopen in $P_n$ and closed in $X$.
$\bf Ques2:$ Is interior of $F_n(x)=\phi?$
Ref: Above picture is from the book 'Fundamentals of General Topology by Arkhangel'ski, p.275.'