{\twoheadrightarrow}Y$, such that $\pi{^{-1}}\{b\}$ is a singleton for some $b\in Y$.

Say $\sigma : X {\twoheadrightarrow}Z$. Fix any $c \in Z$, and then fix $a \in \sigma{^{-1}}\{c\}$ such that $a$ is isolated in $\sigma{^{-1}}\{c\}$. Since $Z$ is metric, $\{a\}$ is a $G_\delta$ in $X$, so fix any $f\in C(X, [0,1])$ with $\{a\} = f{^{-1}}\{1\}$. Choose $\pi\in{{\mathfrak M}_{\mathfrak S}}(X)$ with $\pi \le \sigma$ and $\pi\le f$.

Only a scattered compactum $X$ has the property that *all* maps in ${{\mathfrak M}}(X)$ are tight: If $X$ is not scattered, then $X$ maps onto $[0,1]^2$; if we follow that map by the usual projection onto $[0,1]$, we get a map from $X$ onto $[0,1]$ which is not even ${{\mathfrak c}}$–tight. The *dissipated* compacta have the property that *cofinally* many of these maps are tight:

\[def-dis\] $X$ is *$\kappa$–dissipated* iff $X$ is compact and whenever $g \in {{\mathfrak M}}(X)$, there is a finer $\kappa$–tight $f \in {{\mathfrak M}}(X)$. $X$ is *dissipated* iff $X$ is $2$–dissipated.

So, the $1$–dissipated compacta are the scattered compacta. Metric compacta are dissipated because we can let $f$ be identity map. By Lemma 3.12 of [@KU2]:

\[lemma-local\] For any $\kappa$, the class of $\kappa$–dissipated compacta is a local class.

An easy 