  The mathematical structure of the sheaf of Dedekind real numbers $\RsubD(X)$
for a quantum system is discussed. The algebra of physical qualities is
represented by an $O^{*}$ algebra $\mathcal M$ that acts on a Hilbert space
that carries an irreducible representation of the symmetry group of the system.
$X =\EsubS(\mathcal M)$, the state space for $\mathcal M$, has the weak
topology generated by the functions $ a_{Q}(\cdot)$, defined for $\hat A \in
\mathcal M_{sa} $ and $\forall \hat \rho \in \EsubS(\mathcal M) $, by $
a_{Q}(\hat \rho) = Tr \hat A \hat \rho $. For any open subset $W$ of
$\EsubS(\mathcal M)$, the function $ a_{Q}|_{W}$ is the numerical value of the
quality $\hat A$ defined to the extent $W$. The example of the quantum real
numbers for a single Galilean relativistic particle is given.
