  A recent experiment has reported a nitrogen-doped lutetium hydride acheving a
remarkable Tc of 294 K at just 1 GPa, significantly reducing the required
pressure for obtaining room temperature superconductivity. However, subsequent
experimental and theoretical investigations have encountered difficulties in
replicating these results, leaving the structure of this Lu-H-N compound
shrouded in uncertainty. Here, we propose a stable structure for Lu$_4$H$_7$N
employing first-principles calculations. Our calculations reveal that
Lu$_4$H$_7$N has a Tc of 1.044 K, which can be substantially enhanced to 11.721
K at 150 GPa, due to the increasing electron-phonon coupling (EPC). Notably, we
delve into the nontrivial Z2 band topology of Lu$_4$H$_7$N, featuring
discernible surface states near the Fermi level, and we explore its spin Hall
conductivity characteristics. Furthermore, we find that the electron doping can
enhance the EPC strength and Tc of Lu$_4$H$_7$N, such as the Lu$_4$H$_7$O
structure we predict simulating electron doping for Lu$_4$H$_7$N with an
impressive Tc of 3.837 K. This work demonstrates the coexistence of
superconducting and topological properties in a Lu-H-N system compound, which
holds the promise of guiding the search for novel topological superconducting
materials.
