Is there a structure theorem for nonempty, compact, nowhere dense subsets of the real line? Let $X$ be the set of all nonempty compact nowhere dense subsets of the real line. Is there a theorem that describes the form of the elements of $X$? ### Context For _open_ subsets of the line, such a result is well-known: every open set is the disjoint union of open intervals. But compact sets can be substantially more complicated.

Up to homeomorphism the basic ones are homeomorphic copies of the ordinal space $\alpha+1$ for each $\alpha<\omega_1$, and Cantor sets. Of course $X$ is also closed under finite unions.

Of course a space homeomorphic to one of the countable compact ordinal space can be embedded in a non-obvious way. For example, $\omega^2+1$ can be embedded as follows:

$$f:\omega^2+1\to\Bbb R:\begin{cases} \omega\cdot n\mapsto\frac1{2^n}\\\\\\\ \omega\cdot n+k\mapsto\frac1{2^{n+1}}-\frac1{2^{n+2+k}},&\text{if }k>0\\\\\\\ \omega^2\mapsto 0\;. \end{cases}$$

The resulting set of reals looks schematically like this in its order in $\Bbb R$:

$$\bullet\dots\longrightarrow\bullet\longrightarrow\bullet\longrightarrow\bullet\longrightarrow\bullet\longrightarrow\bullet\bullet$$

The bullets ($\bullet$) from right to left are $f(\omega\cdot0),f(\omega\cdot 1),f(\omega\cdot2),\ldots,f(\omega^2)$. This is rather different from our usual picture of $\omega^2+1$ in its natural (ordinal) order.