  We prove a large deviation principle for the point process associated to
$k$-element connected components in $\mathbb R^d$ with respect to the
connectivity radii $r_n\to\infty$. The random points are generated from a
homogeneous Poisson point process, so that $(r_n)_{n\ge1}$ satisfies
$n^kr_n^{d(k-1)}\to\infty$ and $nr_n^d\to0$ as $n\to\infty$ (i.e., sparse
regime). The rate function for the obtained large deviation principle can be
represented as relative entropy. As an application, we deduce large deviation
principles for various functionals and point processes appearing in stochastic
geometry and topology. As concrete examples of topological invariants, we
consider persistent Betti numbers of geometric complexes and the number of
Morse critical points of the min-type distance function.
