A nondegenerate continuum $X$ is completely regular if each nondegenerate subcontinuum of $X$ has nonempty interior. The class of completely regular continua contains all nontrivial connected finite topological graphs and is contained in the class of all regular curves.

Let $CR(I^n)$ denote the hyperspace of completely regular subcontinua of the cube $I^n$, $2\le n\le\infty$, considered as a subspace of the Vietoris hyperspace $C(I^n)$ of all subcontinua of $I^n$.

We will discuss the descriptive complexity of $CR(I^n)$: the hyperspace is a Borel subset of $C(I^n)$ which is not $F_{\sigma\delta\sigma}$. In fact, in the spirit of the theory of absorbing sets, one can show that $CR(I^n)$ is an absolute retract which is strongly $G_{\delta\sigma\delta}$-universal in the topological Hilbert cube $C(I^n)$.