PomeCorollary 13.27.9.5AF48DFCEE5A The ideal models in $\mathcal M_+(\theta )$ form a closed, hence complete, subspace of $(\mathcal M_+(\theta ),\mathfrak d_\theta )$, and the analytic ordinary models of positive lower volume are dense in it. Equivalently, this subspace is the metric completion of the analytic ordinary models of positive lower volume with the restricted metric.
Corollary 13.27.95AF48DFCEE5A
Proof. By theorem 13.24.5, an ordinary model of positive lower volume that is a $d_\theta $-limit of ideal models is again an ideal model. The same theorem, which relies on theorem 13.24.3, gives the density of analytic models. Now apply (13.27.5) and the completeness of $\mathfrak d_\theta $. $\square$
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