  We show that an inhomogeneous compact extra space possesses two necessary
features --- their existence does not contradict the observable value of the
cosmological constant $\Lambda_4$ in pure $f(R)$ theory, and the extra
dimensions are stable relative to the "radion mode" of perturbations, the only
mode considered. For a two-dimensional extra space, both analytical and
numerical solutions for the metric are found, able to provide a zero or
arbitrarily small $\Lambda_4$. A no-go theorem has also been proved, that
maximally symmetric compact extra spaces are inconsistent with 4D Minkowski
space in the framework of pure $f(R)$ gravity.
