Assume $X$ is compact. Let ${{\mathfrak M}}(X)$, the *metric projections* of $X$, be the class of all maps $\pi$ such that $\pi : X \to Y$ for some compact metric $Y$. Then $\pi \in {{\mathfrak M}_{\mathfrak S}}(X) \subseteq {{\mathfrak M}}(X)$ iff in addition, each $\pi{^{-1}}\{y\}$ is scattered.

\[lemma-MS-down\] If $\pi,\sigma\in{{\mathfrak M}}(X)$ and $\pi \le \sigma \in {{\mathfrak M}_{\mathfrak S}}(X)$, then $\pi \in {{\mathfrak M}_{\mathfrak S}}(X)$.

Observe that in the definition of $f \le g$, it is irrelevant whether $f,g$ map $X$ *onto* $Y,Z$. Here, and in the definition of ${{\mathfrak M}}(X)$, we should really regard $f$ in the set-theoretic sense as a set of ordered pairs, not as a triple $(f,X,Y)$, so that $f: X \to Y$ and $f: X {\twoheadrightarrow}f(X)$ are exactly the same object. One could also define ${{\mathfrak M}}(X)$ and ${{\mathfrak M}_{\mathfrak S}}(X)$ as sets of closed equivalence relations on $X$.

\[lemma-count\] ${{\mathfrak M}}(X)$ is countably directed. That is, if $\sigma_n \in {{\mathfrak M}}(X)$ for $n \in \omega$, then there is a $\pi \in {{\mathfrak M}}(X)$ with $\pi \le \sigma_n$ for each $n$.

\[lemma-sing\] If $\sigma \in {{\mathfrak M}_{\mathfrak S}}(X)$, then there is a $\pi \in {{\mathfrak M}_{\mathfrak S}}(X)$ with $\pi \le \sigma$ and $\pi : X 