PomeProposition 18.12.1.DAA4A0418197 A countable family $\{ F_ k\} _{k\geq 1}\subset C^\infty (X,\mathbb R)$ that is dense for the uniform norm suffices for the obstacle coordinates of $\tau ^{\mathrm{all}}_\omega $. For smooth $\chi $ with $\omega '=\omega +\mathrm{dd}^{\mathrm{c}}\chi >0$, the map $[u]\mapsto [u-\chi ]$ is a uniform isomorphism between $\tau ^{\mathrm{all}}_\omega $ and $\tau ^{\mathrm{all}}_{\omega '}$. For $s>0$, positive scaling gives the corresponding uniform isomorphism onto $\tau ^{\mathrm{all}}_{s\omega }$. This topology is not $T_0$ on ordinary classes.
Proposition 18.12.1DAA4A0418197
Proof. For either envelope operator, the obstacle comparison gives
\[ \| P_{\omega ,F}[u]-P_{\omega ,G}[u]\| _{L^1(\mathrm dV)} \leq \| F-G\| _\infty . \]
Hence the two envelope distances for a pair $[u],[v]$ differ by at most $2\| F-G\| _\infty $, uniformly in the pair. The space $C^\infty (X,\mathbb R)$ contains a countable subset that is dense for the uniform norm: approximate each member of a countable dense subset of $C(X)$ by smooth functions with errors $1/m$, $m\geq 1$. Given an entourage, we approximate each of its finitely many obstacles by a member of the family and shrink the radii accordingly; this proves that the family generates the same uniformity. The divisorial coordinates still range over all fixed compact modifications.
The transport in proposition 18.11.1 relabels the entire family of smooth obstacles by $F\mapsto F-\chi $, or by $F\mapsto sF$. These are bijections of the family, which proves both isomorphisms.
For nonseparation, use the unbounded function $u=1-\sqrt{1-w}$ of proposition 18.10.1. It has trivial multiplier ideals and zero divisorial Lelong numbers, and $\delta w\preceq u$ for $0<\delta \leq 1$. Fix $F$ and put $q=P_\omega (F)$, the envelope of the obstacle $F$ in the sense of section 5.40. Since $\omega >0$, the constant $\min _ XF$ is a competitor, so $q$ is bounded and $q\leq F$. The function
\[ h_\delta =(1-\delta )q+\delta w-\delta \| F\| _\infty \]
is $\omega $-plurisubharmonic, lies below $F$, and satisfies $h_\delta \preceq u$. Letting $\delta \downarrow 0$ off the pole of $w$ gives $P_{\omega ,F}[u]=q=P_{\omega ,F}[0]$. Since $u$ and $0$ have the same multiplier ideals, their ideal envelopes below $F$ also agree. Every generating entourage therefore has identical sections at $[u]$ and $[0]$, although these ordinary classes differ. $\square$
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