The monotone union of open $n$-cells is an open $n$-cell
Morton B. Brown · Proceedings of the American Mathematical Society · 1961
In a research announcement [2] B. Mazur indicated that modulo the Generalized Schoenflies Theorem, the following theorem could be proved: 'If the open cone over a topological space X is locally Euclidean at the origin, then it is topologically equivalent with Euclidean space. Ronald Rosen [3 ] has described an ingenious proof of this theorem based on the now known [1 ] Generalized Schoenflies Theorem. In the present paper we prove a stronger theorem without employing the Generalized Schoenflies Theorem. DEFINITIONS AND NOTATION. If Q is an n-cell then 4, Q denote the interior and boundary of Q, respectively. An n-annulus is a homeomorph of Sn-1 X [01]. If S is an (n 1)-sphere in an n-cell, then I(S) denotes the interior (complementary domain) of S. If SI, S2 are (n-1)-spheres in an n-cell and S, CI(S2), then [S1, S2] (or equivalently [S2, S1]) denotes the set Cl [I(S2) ] -I(SI). An (n1)-sphere S embedded in a space X is collared if there is a homeomorphism h of Sn-I X [01o into X such that h(Sn-1 X 1/2) = S. Finally B, will denote the n-ball of radius r in En and centered at the origin.