M}_{\mathfrak S}}(X) \ne \emptyset$.

\[lemma-clh\] The class of $\kappa$–superdissipated compacta is a local class.

By Lemma \[lemma-tight-LOTS\]:

The standard map $\sigma : I_\Lambda {\twoheadrightarrow}I$ is $2$–supertight.

The situation for products is more complicated. By Lemma \[lemma-tight-prod\] and induction:

\[lemma-big-lim\] For any $n \ge 1$ and $S_i \subseteq I $ (for $i < n$): The standard map $\sigma : \prod_{i <n} I_{S_i} {\twoheadrightarrow}I^n$ is $(2^{n-1} + 1)$–supertight.

This result is best possible by Theorem 3.9 of [@KU2]; a product $\prod_{i< n}X_i$ is not $(2^{n-1} )$–dissipated if each $X_i$ is a compact separable LOTS, none of the $X_i$ is scattered, and at most one of the $X_i$ is second countable.

\[def-kernel\] The *perfect kernel*, $\ker(X)$, is $\emptyset$ if $X$ is scattered, and the largest perfect subset of $X$ otherwise.

By Lemma \[lemma-tight-equiv\], the tightness of $\pi : X \to Y$ can be expressed using perfect subsets of $X$, so that

\[lemma-ker-tight\] $\pi : X \to Y$ is $\kappa$–(super)tight iff $\pi{\mathord {\upharpoonright}}\ker(X)$ is $\kappa$–(super)tight, and the space $X$ is $\kappa$–(super)dissipated iff $\ker(X)$ is $\kappa$–(super)dissipated.

\[lemma-get-one\] Assume that $\pi: X \to Y$ is $(n+2)$–supertight, where $n \in \omega$, $X$ is compact and $Y$ is compact metric, and $\{P_0, \ldots, P_n\}$ is a loose family 