  The Lusternik-Schnirelmann category $cat(X)$ is a homotopy invariant which is
a numerical bound on the number of critical points of a smooth function on a
manifold. Another similar invariant is the topological complexity $TC(X)$ (a la
Farber) which has interesting applications in Robotics, specifically, in the
robot motion planning problem. In this paper we calculate the
Lusternik-Schnirelmann category and as a consequence we calculate the
topological complexity of the two-point ordered configuration space of
$\mathbb{CP}^n$ for every $n\geq 1$.
