Locally flat strings and half-strings
C. Lacher · Proceedings of the American Mathematical Society · 1967
1. Definitions. Rn denotes euclidean n-space, Hn = Rn-X [0, oo) CRn, Bn the unit ball in Rn, and Sn the one-point compactification of Rn. An n-string, n-half-string, n-cell, n-sphere, a set which homeomorphic to Rn, Hn, Bn, Sn respectively. An n-manifold a space M such that each point of M has a neighborhood homeomorphic to Rn; an n-manifold with boundary a space each point of which has a neighborhood whose closure an n-cell. If M an n-manifold with boundary, the set of points of M which have neighborhoods homeomorphic to Rn denoted by M[ (the interior of M) and MA denoted by M (the boundary of M). Let M be a k-manifold with boundary contained in the n-manifold N; M locally flat at the point pClA if there a neighborhood U of p in N such that (U, UnM) homeomorphic to the pair (Rn, Rk); M locally flat at the point pEk if there a neighborhood U of p in N such that (U, UnM) homeomorphic to (Rn, Hk). The symbol ; will be used to mean is homeomorphic to.