  A critical point is an important structure in the phase diagram of a
thermodynamic system. In this work, we introduce topology to the study of the
black hole thermodynamics for the first time by following Duan's topological
current $\phi$-mapping theory. Each critical point is endowed with a
topological charge. We find that critical points can be divided into two
classes, the conventional and the novel. Further study shows that the
first-order phase transition can extend from the conventional critical point,
while the presence of the novel critical point cannot serve as an indicator of
the presence of the first-order phase transition near it. Moreover, the charged
anti-de Sitter black hole and the Born-Infeld anti-de Sitter black hole have
different topological charges, which indicates they are in different
topological classes from the viewpoint of thermodynamics. These give the first
promising study on the topology of the black hole thermodynamics. Such approach
is also expected to be extended to other black holes, and much more topological
information remains to be disclosed.
