Clopen boxes and the P-space property

For $a(n)<b(n)\le {\mathrm{\aleph }}_n$, F3 makes $(a(n),b(n)]$ open in the factor $[0,{\mathrm{\aleph }}_n]$. Its complement is $[0,a(n)]\cup (b(n),{\mathrm{\aleph }}_n]$; the first is basic open, and the second is basic open if $b(n)<{\mathrm{\aleph }}_n$ and empty otherwise. Thus the coordinate interval is also closed. The box ${\prod }_{n\in B}(a(n),b(n)]$ is open by the definition of the box topology. Its complement is the union over $n$ of cylinders restricting just that coordinate to its open complement, with all other factors unrestricted, and hence is box-open. Intersecting with either $Z$ in F1–F2 shows $(a,b{]}_Z$ is clopen, including the possibility it is empty. No condition on the cofinalities of $a$ or $b$ was used.

Let $h\in Z$ and let $U$ be an open neighborhood of $h$. By F1–F2 choose an open factor box $\prod O_n$ with $h\in Z\cap \prod O_n\subseteq U$. Each $h(n)$ has uncountable cofinality by F2, hence is a nonzero limit ordinal. F3 supplies some $a(n)<h(n)$ with $(a(n),h(n)]\subseteq O_n$. One may take the least such ordinal separately at each coordinate, so these lower bounds form a specified $a\in P_B$. Now $h\in (a,h{]}_Z\subseteq U$, and step 1.1 makes this box clopen. This proves the local-base assertion, including at coordinate tops $h(n)={\mathrm{\aleph }}_n$.

Let $(U_j{)}_{j<\omega }$ be open in $Z$ and let $h\in {\bigcap }_j{U}_j$. By step 2.1 each $U_j$ has a nonempty set of lower bounds $a_j<h$ whose local boxes lie in $U_j$. Apply A1 to select them. Put $a(n)={\sup }_{j<\omega }{a}_j(n)$. By F2 and F4 this is strictly below $h(n)$ at every coordinate, so $a\in P_B$ and $a<h$. For $x\in (a,h{]}_Z$, the inequalities $a_j(n)\le a(n)<x(n)\le h(n)$ place $x$ in every $(a_j,h{]}_Z\subseteq U_j$. Thus step 1.1 gives an open neighborhood $(a,h{]}_Z$ of $h$ inside the intersection. Every point in the intersection has such a neighborhood, so it is open; if ${\bigcap }_j{U}_j$ has no points, it is the empty open set. Finite nonempty families reduce to this case by adding whole-space terms, and the intersection of a family with no members is $Z$, also open. Only individual coordinate cofinalities were used, so the argument applies to both spaces without any uniform bound. QED.