  In this paper we investigate the $(2+1)$-dimensional topological non-Fermi
liquid in strongly correlated electron system, which has a holographic dual
description by Einstein gravity in $(3+1)$-dimensional anti-de Sitter (AdS)
space-time. In a dyonic Reissner-Nordstrom black hole background, we consider a
Dirac fermion coupled to the background $U(1)$ gauge theory and an intrinsic
chiral gauge field $b_M$ induced by chiral anomaly. UV retarded Green's
function of the charged fermion in the UV boundary from AdS$_4$ gravity is
calculated, by imposing in-falling wave condition at the horizon. We also
obtain IR correlation function of the charged fermion at the IR boundary
arising from the near horizon geometry of the topological black hole with index
$k=0,\pm 1$. By using the UV retarded Green's function and IR correlation
function, we analyze the low frequency behavior of the topological non-Fermi
liquid at zero and finite temperatures, especially the relevant non-Fermi
liquid behavior near the quantum critical region. In addition, we find that
when $k=\pm 1$, the effective mass of bulk Dirac fermion at IR fixed point is
topologically quantized, consequently the Fermi momentum presents a discrete
sparse distributed pattern, in contrary to the Ricci flat case with $k=0$. As
demonstrated examples, we calculate the spectral functions of topological
non-Fermi liquid due to anomalous Hall effect (AHE) ($|\vec{b}|\ne 0, b_0=0$)
and chiral magnetic effect (CME) ($|\vec{b}| = 0, b_0\ne 0$), which can be
tested in topological materials such as topological insulator and topological
metal with strong couplings.
