A local cut point is a point
that disconnects its sufficiently small neighborhood. We show that there
exists an upper bound for the degree of a local cut point in a metric measure
space satisfying the generalized Bishop–Gromov inequality. As a corollary, we
obtain an upper bound for the number of ends of such a space. We also
obtain some obstruction conditions for the existence of a local cut point
in a metric measure space satisfying the Bishop–Gromov inequality or the
Poincaré inequality. For example, the measured Gromov–Hausdorff limits of
Riemannian manifolds with a lower Ricci curvature bound satisfy these two
inequalities.
