1. Introduction. Although Corollaries 4 and 5 to Theorem 1 below appear elsewhere in the literature (1, 2), the proofs given seem to use rather long and involved arguments and refer to other results in the literature for their completeness. The proofs given below are brief and follow quite naturally in sequence with the other corollaries from Theorem 1. The arguments presented are independent of references to the literature except for the reference in the proof of Theorem 1 to Lemma 2·1 of (3), which is well known to topologists. For our purposes this theorem may be stated as: For each open 2-manifold M (non-compact and without boundary), there is a subcomplex L made up of edges of some triangulation of M such that the open simplicial neighbourhood of L is piecewise linearly homeomorphic to M.