  Given a real analytic (or, more generally, semianalytic) set R in the
n-dimensional complex space, there is, for every point p in the closure of R, a
unique smallest complex analytic germ X_p that contains the germ R_p. We call
the complex dimension of X_p the holomorphic closure dimension of R at p. We
show that the holomorphic closure dimension of an irreducible R is constant on
the complement of a closed proper analytic subset of R, and discuss the
relationship between this dimension and the CR dimension of R.
