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 $X$?
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.
It is the Mazurkiewicz–Sierpiński theorem combined with the fact that every compact perfect metric space contains a copy of the Cantor set. So in your case, the Mazurkiewicz–Sierpiński theorem applies and all countable, successor ordinals exhaust the homeomorphism type of non-empty compact nowhere-dense subsets of the real line. (All countable ordinals embed into $[0,1]$.)
Stefan Mazurkiewicz and Wacław Sierpiński, Contribution à la topologie des ensembles dénombrables, Fundamenta Mathematicae 1 (1920), 17–27.
It is noteworthy that this theorem appeared in the very first volume of Fundamenta Mathematicae.
