Let $X$ be a (perfectly normal) topological space.
There is a family $\{C_i:i∈I\}$ indexed with an uncountable linear order $(I,\prec)$ such that $Ci\subset Cj$ whenever $i\prec j$. The boundaries of $C_i$ and $C_j$ does not intersect.
Condition 1. For any family defined above, there exists a "volume measure", which is a (continuous) real-valued function, $V(\cdot)$, such that $C_i\subset C_j\iff V(C_i)<V(C_j)$.
Question. What are some necessary and/or sufficient topological conditions on $X$ of Condition 1?
For examples: one sufficient condition is $X$ being a metric space. To see this, let $1\in I$, then, for all $j$ define $V(C_j)=d(bd(C_j),C_1)-d(bd(C_1),C_j)$.
One necessary condition seems to be $T_6$.