  The question of the global topology of the Universe (cosmic topology) is
still open. In the $\Lambda$CDM concordance model it is assumed that the space
of the Universe possesses the trivial topology of $\mathbb{R}^3$ and thus that
the Universe has an infinite volume. As an alternative, we study in this paper
one of the simplest non-trivial topologies given by a cubic 3-torus describing
a universe with a finite volume. To probe cosmic topology, we analyse certain
structure properties in the cosmic microwave background (CMB) using Betti
Functionals and the Euler Characteristic evaluated on excursions sets, which
possess a simple geometrical interpretation. Since the CMB temperature
fluctuations $\delta T$ are observed on the sphere $\mathbb{S}^2$ surrounding
the observer, there are only three Betti functionals $\beta_k(\nu)$, $k=1,2,3$.
Here $\nu=\delta T/\sigma_0$ denotes the temperature threshold normalized by
the standard deviation $\sigma_0$ of $\delta T$. Analytic approximations of the
Gaussian expectations for the Betti functionals and an exact formula for the
Euler characteristic are given. It is shown that the amplitudes of
$\beta_0(\nu)$ and $\beta_1(\nu)$ decrease with increasing volume $V=L^3$ of
the cubic 3-torus universe. Since the computation of the $\beta_k$'s from
observational sky maps is hindered due to the presence of masks, we suggest a
method yielding lower and upper bounds for them and apply it to four Planck
2018 sky maps. It is found that the $\beta_k$'s of the Planck maps lie between
those of the torus universes with side-lengths $L=2.0$ and $L=3.0$ in units of
the Hubble length and above the infinite $\Lambda$CDM case. These results give
a further hint that the Universe has a non-trivial topology.
