Approximation of imbeddings of manifolds in codimension one

It is shown that any $(n-1)$-manifold topologically imbedded in a Euclidean space of dimension greater than four can be approximated arbitrarily closely by one whose complement has the property of uniform local one-connectedness.
From this theorem and the results of Chernavskii and Kirby–Siebenmann it is deduced that there also exists a piecewise linear approximation if the dimension of the Euclidean space is greater than five.
Bibliography: 15 titles.