  We develop a theory of point defects in cholesterics and textures in
spherical droplets with normal anchoring. The local structure of chiral defects
is described by singularity theory and a smectic-like gradient field
establishing a nexus between cholesterics and smectics mediated by their
defects. We identify the defects of degree $-2$ and $-3$ observed
experimentally with the singularities $D_4^{-}$ and $T_{4,4,4}$, respectively.
Radial point defects typical of nematics cannot be perturbed into chiral
structures with a single handedness by general topological considerations. For
the same reasons, the spherical surface frustrates the chirality in a surface
boundary layer containing regions of both handedness.
