  In this paper, via employing the uniformly modified form of the generalized
off-shell Helmholtz free energy, we investigate the topological numbers for the
four-dimensional neutral Lorentzian Taub-NUT, Taub-NUT-AdS and Kerr-NUT
spacetimes, and find that these solutions can also be classified into one of
three types of those well-known black hole solutions, which implies that these
spacetimes should be viewed as generic black holes from the viewpoint of the
thermodynamic topological approach.
