A ZFC Dowker space of size aleph omega plus one

Let $(H_j{)}_{j\in J}$ be an indexed discrete family of closed subsets of $X$. Since $X$ is closed in $R$ by F1, each $H_j$ is closed in $R$: write it as $X\cap C_j$ for a closed $C_j$ in $R$. At a point of $X$, a relative open neighborhood meeting at most one indexed $H_j$ is the intersection with $X$ of an ambient open set. That ambient set meets precisely the same members, since every $H_j$ lies in $X$. At a point outside $X$, the open set $R\setminus X$ meets none. Thus the family is indexed discrete in $R$. F3 under A1 gives pairwise disjoint open neighborhoods $V_j$ there, and $V_j\cap X$ are pairwise disjoint open neighborhoods of $H_j$ in $X$. Empty members may receive the empty set, and an empty index set gives the empty separating family.

Put $E_k=F_k\cap X$. F4 implies these are closed in $X$, decrease, and have empty intersection. Let $(U_k)$ be any open expansion sequence in $X$, with $E_k\subseteq U_k$. Choose open $V_k\subseteq R$ satisfying $V_k\cap X=U_k$, using A1 if necessary, and set $W_k=V_k\cup (R\setminus X)$. This is open by F1. Every point of $F_k$ outside $X$ is in its second summand; every point of $F_k$ inside $X$ belongs to $E_k\subseteq U_k\subseteq V_k$. Therefore $F_k\subseteq W_k$, and $W_k\cap X=U_k$. F4 supplies a bound $b$ whose entire $R$-tail lies in every $W_k$. F2 gives an actual $x\in X$ with $b<x$ pointwise. Thus $x\in W_k\cap X=U_k$ for every $k$, proving ${\bigcap }_k{U}_k\mathrm{\ne }\mathrm{\varnothing }$.

Hausdorff separation in $R$ restricts to separation in $X$ by intersecting the two disjoint ambient neighborhoods with $X$. Consequently $X$ is $T_1$: for fixed $x$, each other point has an open neighborhood avoiding $x$, and their union is $X\setminus \{x\}$. For disjoint closed $C,D\subseteq X$, their two-member family is discrete: the complement of $D$ works at points of $C$, the complement of $C$ at points of $D$, and the complement of their union elsewhere. Step 1.1 separates this family. Thus $X$ is normal as well as Hausdorff and collectionwise normal.

If $X$ were countably paracompact, its normality from step 2.1 and F5 would give open expansions of the sequence $(E_k)$ with empty intersection. Step 1.2 excludes every such sequence. Hence $X$ is not countably paracompact; together with step 2.1 and F6, this makes $X$ a Dowker space.

Closedness is F1, and F2 gives both strict pointwise cofinality and the cardinality ${\mathrm{\aleph }}_{\omega +1}$. Steps 1.1 and 2.1 establish the stated separation properties, and step 3.1 establishes the Dowker conclusion. No inference that failure of countable paracompactness passes to arbitrary closed subspaces is needed: step 1.2 proves the required failure for this particular cofinal subspace. QED.