P}}{\sqsubseteq}C({{\mathbb T}})$; note that the restriction ${{\mathcal P}}{\mathord {\upharpoonright}}A {\sqsubseteq}C(A)$ is dense in $C(A)$ by Mergelyan’s Theorem.

Ordered Spaces {#sec-LOTS}
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We begin by defining the double arrow space and some variants thereof:

\[def-das\] $I = [0,1]$. If $\Lambda : I \to \omega$, then $I_\Lambda = \bigcup_{x \in I} \{x\} \times \{0,1,\ldots, \Lambda(x)\}$, which is given the lexicographic order and the usual order topology. If $S \subseteq (0,1)$, then $I_S  = I_{{{\raise 2 pt \hbox{$\chi$}}}_S}$, where ${{\raise 2 pt \hbox{$\chi$}}}_S$ is the characteristic function; then for $x \in S$, let $x^- = (x,0)$ and $x^+ = (x,1)$; while if $x \notin S$, let $x^- = x^+ = (x,0)$. The *double arrow space* is $I_{(0,1)}$. For any $\Lambda$, the map $(x, \ell) \mapsto x$ is the *standard map* from $I_\Lambda$ onto $I$.

So, we form $I_\Lambda$ by splitting each $x \in S$ into $\Lambda(x) + 1$ neighboring points. For $I_S$, we split each $x \in S$ into two neighboring points, $x^-,x^+$, and we don’t split the points in $I \backslash S$; it is convenient to have $x^\pm$ defined for *all* $x \in I$, so, for example, we can say that for all $a < b$ in $I$, $(a^+, b^-)$ is an 