double arrow space onto $[0,1]$, which is tight by Lemma 2.3 of [@KU2]:

\[lemma-tight-LOTS\] If $X,Y$ are compact LOTSes and $f:X \to Y$ is order-preserving $(x_1 < x_2 \to f(x_1) \le f(x_2))$, then $f$ is tight.

One can estimate the tightness of product maps using Lemma 2.14 of [@KU2]:

\[lemma-tight-prod\] Assume that for $i = 0,1$: $X_i, Y_i$ are compact, $f_i:X_i \to Y_i$ is $(m_i + 1)$–tight, $m_i \le n_i < \omega$, and $|f_i{^{-1}}\{y\}| \le n_i$ for *all* $y \in Y_i$. Then $f_0 \times f_1 : X_0 \times X_1 \to Y_0 \times Y_1$ is $(\max(m_0 n_1 , m_1 n_0) + 1)$– tight.

The notion of a *dissipated* compactum (Definition \[def-dis\] below) involves tight maps onto metric compacta, ordered by *fineness*, so we define:

\[def-fine\] Assume that $X,Y,Z$ are compact, $f : X \to Y$, and $g : X \to Z$. Then $f \le g$, or $f$ is *finer than* $g$, iff there is a $\Gamma \in C(f(X),g(X))$ such that $g = \Gamma \circ f$.

\[lemma-fine-equiv\] Assume that $X,Y,Z$ are compact, $f : X \to Y$, and $g : X \to Z$. Then $f \le g$ iff $\; \forall x_1, x_2 \in X\, [ f(x_1) = f(x_2) \to g(x_1) = g(x_2)]$.

