  The quest for a consistent theory for quantum gravity is one of the most
challenging problems in theoretical high-energy physics. An often-used approach
is to describe the gravitational degrees of freedom by the metric tensor or
related variables, and finding a way to quantise this. In the canonical tensor
model, the gravitational degrees of freedom are encoded in a tensorial quantity
$P_{abc}$, and this quantity is subsequently quantised. This makes the
quantisation much more straightforward mathematically, but the interpretation
of this tensor as a spacetime is less evident. In this work we take a first
step towards fully understanding the relationship to spacetime. By considering
$P_{abc}$ as the generator of an algebra of functions, we first describe how we
can recover the topology and the measure of a compact Riemannian manifold.
Using the tensor rank decomposition, we then generalise this principle in order
to have a well-defined notion of the topology and geometry for a large class of
tensors $P_{abc}$. We provide some examples of the emergence of a topology and
measure of both exact and perturbed Riemannian manifolds, and of a purely
algebraically-defined space called the semi-local circle.
