A topological space X is said to have the disappearing closed set (DCS) property if and only if for every proper closed subset C there is a sequence of homeomorphisms {hi}, i = 1, 2, 3, * * * of X onto X, and a decreasing sequence of open subsets {Uif, i=1, 2, 3, *, of X such that nl?oUi=0 and hi(C) Ui. THEOREM. A finite simplicial n-conmplex is an n-manifold if and only if it has the DCS property. A topological space X is said to have the disappearing closed set (DCS) property if and only if for every proper closed set CcX there is a decreasing family of open sets {U}, i= 1, 2, * , in Xsuch that nfl 1 Ui= 0 and a sequence of homeomorphisms {hi}j=1 of X onto X such that hi(C)c Ui, i=1, 21,* This definition was motivated by an attempt to study ideas related to invertible spaces ([1], [2]). Examples of spaces with the DCS property are the n-sphere, open n-cell, torus and open annulus. A disconnected example is (0, 1)n{rationals}. It can be proved that any disconnected DCS space must have infinitely many components, and that the product of two DCS spaces will also have the DCS property [3]. It is proved here that, under certain restrictions, the union of two spaces each of which has the DCS property will have the DCS property. This is then used to prove that a finite simplicial ui-complex is an n-manifold if and only if it has the DCS property. THEOREM 1. Let X be a space wvith two intersecting open subspaces C and D each of which has the DCS property, such that for any proper closed subset of C (or D, resp.), the sequences of open sets {Ui} l ({Vi} 1) and homeomorphisms {h}[=1 ({ki}j? 1) may be taken to have the following properties: (a) hi (ki) may be extended to a homeomorphism of X onto X that is the identity on X-C (X-D). (b) T1here is a positive integer M (N) such that Cr(DU_1)$ 0 (D r (Cr\) o0 ). (The closure is with respect to C U D.) Presented to the Society, April 10, 1971; received by the editors March 3, 1971. AMS 1970 subject classifications. Primary 54D99, 57A05, 57AlO, 57A14.