$| \widehat f _y | < n$, and hence $\Phi((  \widehat f _y )^{n}) = \{0\}$, for all but countably many $y$ (see Lemma \[lemma-tight-equiv\]). But for all $y$, $\pi{^{-1}}\{y\}$ is scattered, so that $\Phi( ( \widehat f _y )^{n})$ is also scattered, and hence countable. Thus, the union of all these sets is also countable.

Dissipation is a notion of smallness, which is balanced by a notion of bigness, which is really a partition property:

\[def-big\] Fix a real $r > 0$. The compact space $X$ is $n$–*big* iff for all ${{\mathcal A}}{\sqsubseteq}C(X)$ and all partitions $\Upsilon : {{\mathcal A}}\to \omega$, there are $f_1, \ldots, f_n \in {{\mathcal A}}$ and a point $c \in X$ such that the $\Upsilon(f_j)$, for $j = 1, \ldots, n$, are all equal, and such that $|f_i(c) - f_j(c)| \ge r$ whenever $1 \le i < j \le n$.

Since ${{\mathcal A}}$ is a linear subspace, it does not matter which $r > 0$ we use. The notion of $1$–big is trivial, and $2$–big is easily characterized:

\[lemma-two-big\] The compact space $X$ is $2$–big iff $X$ is not second countable.

Note that $\exists c \,\big(|f_1(c) - f_2(c)| \ge r\big)$ holds iff $\|f_1 - f_2\| 