  It was discovered some years ago that there exist non-integer real numbers
$q>1$ for which only one sequence $(c_i)$ of integers $c_i \in [0,q)$ satisfies
the equality $\sum_{i=1}^\infty c_iq^{-i}=1$. The set of such "univoque
numbers" has a rich topological structure, and its study revealed a number of
unexpected connections with measure theory, fractals, ergodic theory and
Diophantine approximation.
  In this paper we consider for each fixed $q>1$ the set $\mathcal{U}_q$ of
real numbers $x$ having a unique representation of the form $\sum_{i=1}^\infty
c_iq^{-i}=x$ with integers $c_i$ belonging to $[0,q)$. We carry out a detailed
topological study of these sets. For instance, we characterize their closures,
and we determine those bases $q$ for which $\mathcal{U}_q$ is closed or even a
Cantor set. We also study the set $\mathcal{U}_q'$ consisting of all sequences
$(c_i)$ of integers $c_i \in [0,q)$ such that $\sum_{i=1}^{\infty} c_i q^{-i}
\in \mathcal{U}_q$. We determine the numbers $r >1$ for which the map $q
\mapsto \mathcal{U}_q'$ (defined on $(1, \infty)$) is constant in a
neighborhood of $r$ and the numbers $q >1$ for which $\mathcal{U}_q'$ is a
subshift or a subshift of finite type.
