  We analyze a correlated system in equilibrium with special emphasis on
non-Hermitian topology inducing a skin effect. The pseudo-spectrum, computed by
the real-space dynamical mean-field theory, elucidates that additional
pseudo-eigenstates emerge for the open boundary condition in contrast to the
dependence of the density of states on the boundary condition. We further
discuss how the line-gap topology, another type of non-Hermitian topology,
affects the pseudo-spectrum. Our numerical simulation clarifies that while the
damping of the quasi-particles induces the non-trivial point-gap topology, it
destroys the non-trivial line-gap topology. The above two effects are also
reflected in the temperature dependence of the local pseudo-spectral weight.
