space obtained from $X$ by replacing each isolated point by a copy of the double arrow space.

Note that $(I_S)^2$ is separable, but it is not hereditarily separable when $S$ is uncountable; in fact, more general $I_\Lambda$ occur naturally in such products. Fixing an uncountable $S \subseteq (0,1)$, let $L_n = I_{\Lambda_n}$, where $\Lambda_n(x) = n$ for $n \in S$, and $\Lambda_n(x) = 0$ for $n \notin S$. Then $L_n$ is not separable whenever $n \ge 2$, and the diagonal of $(I_S)^k$ is homeomorphic to $L_{2^k -1}$.

Tight Maps and Dissipated Spaces {#sec-tight}
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We recall some definitions and results from [@KU2]. As usual, $f : X \to Y$ means that $f$ is a *continuous* map from $X$ to $Y$, and $f : X {\twoheadrightarrow}Y$ means that $f$ is a continuous map from $X$ *onto* $Y$.

\[def-tight\] Assume that $X,Y$ are compact and $f: X \to Y$.

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A *loose family* for $f$ is a disjoint family ${{\mathcal P}}$ of closed subsets of $X$ such that for some non-scattered $Q \subseteq Y$, $Q = f(P)$ for all $P \in {{\mathcal P}}$.

$f$ is $\kappa$–*tight* iff there are no loose families for $f$ of size $\kappa$.

$f$ is *tight* iff $f$ 