  The Statistical Learning Theory (SLT) provides the foundation to ensure that
a supervised algorithm generalizes the mapping $f: \mathcal{X} \to \mathcal{Y}$
given $f$ is selected from its search space bias $\mathcal{F}$. SLT depends on
the Shattering coefficient function $\mathcal{N}(\mathcal{F},n)$ to upper bound
the empirical risk minimization principle, from which one can estimate the
necessary training sample size to ensure the probabilistic learning convergence
and, most importantly, the characterization of the capacity of $\mathcal{F}$,
including its underfitting and overfitting abilities while addressing specific
target problems. However, the analytical solution of the Shattering coefficient
is still an open problem since the first studies by Vapnik and Chervonenkis in
$1962$, which we address on specific datasets, in this paper, by employing
equivalence relations from Topology, data separability results by Har-Peled and
Jones, and combinatorics. Our approach computes the Shattering coefficient for
both binary and multi-class datasets, leading to the following additional
contributions: (i) the estimation of the required number of hyperplanes in the
worst and best-case classification scenarios and the respective $\Omega$ and
$O$ complexities; (ii) the estimation of the training sample sizes required to
ensure supervised learning; and (iii) the comparison of dataset embeddings,
once they (re)organize samples into some new space configuration. All results
introduced and discussed along this paper are supported by the R package
shattering (https://cran.r-project.org/web/packages/shattering).
