In part 6) of example 103 in Steen & Seebach's "Counterexamples in Topology" they outline a proof that uncountable products of $\mathbb{Z}$ ($X_{\lambda}=\prod_{\alpha\in A}Z^+_{\alpha}$) are not normal by trying to show that any neighbourhoods U & V of $P_0=\{x:\text{only 0 may appear more than once among the coordinates of x}\}$ & $P_1=\{x:\text{only 1 may appear more than once among the coordinates of x}\}$ must intersect. They define a sequence of finite index sets $F_n\subset A$ and a sequence of points $x^n\in X_{\lambda}$, and define $F_n(x^n)=\cap_{\alpha\in F_n}\pi_{\alpha}^{-1}(x^n)$ (i.e. the set of all points in $X_{\lambda}$ that are the same as $x^n$ on the indices in $F_n$). Then they say that $F_n(x^n)\subset U$. WHY?
Surely $F_n(x^n)$ may contain points with any non-zero integer in indices outside of $F_n$, and those numbers may repeat which would exclude the point from $U$. What am I misunderstanding here?
