this is proved by induction in "k" using the mayer–vietoris sequence. when "n" = "k", the zeroth reduced homology of "y" has rank 1, which means that "y" has 2 connected components (which are, moreover, path connected), and with a bit of extra work, one shows that their common boundary is "x". a further generalization was found by j. w. alexander, who established the alexander duality between the reduced homology of a compact subset "x" of r and the reduced cohomology of its complement. if "x" is an "n"-dimensional compact connected submanifold of r (or s) without boundary, its complement has 2 connected components.