Relative consistency of DC with failure of Stone's theorem

Invoke the external relative-consistency construction of [F5], in its regular-$\lambda$ form with $\lambda ={\omega }_1$, and write $N$ for its symmetric model. The invocation is exactly the source's relative-consistency theorem; it does not infer a transitive ground model from $\mathrm{Con}(\mathrm{Z}\mathrm{F})$.

The model satisfies DC by [F2], because every $\omega$-sequence in the full extension with values in $N$ is already in $N$ by [F1].

In $N$ form the metric sum $X:={⨆}_{\xi <\lambda }{R}_{\xi }$ with the metric of [F4]; it is a metrizable space whose components $R_{\xi }$ are clopen and connected.

Let $\mathcal{U}=\{B_X(x,1\mathrm{/}3):x\in X\}$. This is an open cover. Every member lies in one component, since distinct components have distance $1$, and is a proper subset of that component: under $d(r,s)=\mathrm{\mid }r-s\mathrm{\mid }\mathrm{/}(1+\mathrm{\mid }r-s\mathrm{\mid })$, a radius-$1\mathrm{/}3$ ball corresponds to a bounded ordinary interval.

Suppose that $\mathcal{U}$ has a locally finite open refining cover $\mathcal{V}$, and define $S=X\setminus {\bigcup }_{V\in \mathcal{V}}(\bar{V}\setminus V),S_{\xi }=S\cap R_{\xi }\mathrm{.}$ Each $S_{\xi }$ is nonempty. Indeed, a point of $R_{\xi }$ has a nonempty open neighbourhood $W\subseteq R_{\xi }$ meeting only finitely many refinement members $V_0,\dots ,V_k$. Starting with $W_0=W$, recursively choose the nonempty open set $W_{i+1}=W_i\cap V_i$ if that intersection is nonempty, and otherwise choose $W_{i+1}=W_i\setminus \overline{V_i}$. The latter is nonempty: an open $W_i$ disjoint from the open set $V_i$ cannot be contained in $\overline{V_i}$. Thus every point of $W_{k+1}$ avoids the boundaries of the $V_i$, and it avoids the boundary of every other refinement member because that member misses $W$. Hence $W_{k+1}\subseteq S_{\xi }$. The set $S_{\xi }$ is also proper. Choose a nonempty $V\in \mathcal{V}$ meeting $R_{\xi }$. Such a member exists because $\mathcal{V}$ covers $R_{\xi }$. By refinement and step 3.1, $V$ lies in $R_{\xi }$ and is contained in a proper ball there. It is therefore a nonempty proper open subset of the connected space $R_{\xi }$, so $\bar{V}\setminus V$ is nonempty and disjoint from $S_{\xi }$. Consequently $\xi \mapsto S_{\xi }$ is a function assigning a nonempty proper subset to every component.

By [F3] no such function exists in $N$; hence $\mathcal{U}$ has no locally finite open refining cover, and $X$ is not paracompact.

Steps 2.1 and 5.1 identify the DC and nonparacompactness conclusions inside the published model. By [F5], that construction has exactly the asserted consistency strength relative to ZF. The argument uses the regular-$\lambda$ presentation and not the finite-support $\omega$-indexed one.