Properties and applications of the distance functions on open sets of the Euclidean space

For an open subset of the Euclidean space of dimension $n$ we consider interior and exterior approximations by sequences of open sets. We prove convergence everywhere of the corresponding sequences of distance functions from boundary as well as convergence almost everywhere for their gradients. As applications we obtain several new Hardy-type inequalities that contain the scalar product of gradients of test functions and the gradient of the distance function from the boundary of an open subset of the Euclidean space.