  In this paper, we examine the role played by topology, and some specific
boundary conditions as well, on the physics of a higher-dimensional black hole.
We analyze the line element of a five-dimensional non-extremal
Reissner-Nordstr\"{o}m black hole to obtain a new family of subspaces that are
types of strong retractions and deformations, and then we extend these results
to higher dimensions in order to deduce the relationship between various types
of transformations. We also study the scalar field perturbations in the
background under consideration and obtain an analytical expression for the
quasibound state frequencies by using the Vieira-Bezerra-Kokkotas approach,
which uses the polynomial conditions of the general Heun functions, and then we
discuss the stability of the system and present the radial eigenfunctions. Our
main goal is to discuss the physical meaning of these mathematical applications
in such higher-dimensional effective metric.
