  Fix a smooth closed manifold $M$. Let $R_M$ denote the space of all pairs
$(g,L)$ such that $g$ is a $C^3$ Riemannian metric on $M$ and the real number
$L$ is not the length of any closed $g$-geodesics. A locally constant geodesic
count function $\pi_M:R_M\rightarrow Z$ is constructed. For this purpose, the
weight of compact open subsets of the space of closed $g$-geodesics is defined
and investigated for an arbitrary Riemannian metric $g$.
