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D . . c“ . . d‘l . , where the dots denote the possible occurrence of other letters). To prove this assertion, assume that the edges labeled c are not sepa- rated by any other pair of the first kind. 20. Here A and B each designate a whole sequence of edges. The important point is that any edge in A must be identified with another edge in A, and similarly for B. No edge in A is to be identified with an edge in B. But this contradicts the fact that the initial and final vertices of either edge labeled “c” are to be identified, in view of step number three.

The fact that the set of all the triangles with v as a vertex can be divided into several disjoint subsets, such that the triangles in each subset can be arranged in cyclic order as described, is an easy consequence of condition (1). However, if there were more than one such subset, then the requirement that 22 have a neighborhood homeomorphic to U2 would be violated. We shall not attempt a rigorous proof of this last assertion. 1 Let S be a compact surface. 1 by prov- ing that S is homeomorphic to a polygon with the edges identified in pairs as indicated by one of the symbols listed at the end of Section 5.

It is readily seen that the set of interior points is an open everywhere dense subset; hence, the set of boundary points is a closed set. The set of boundary points of an n—dimensional manifold is an (n — 1)-dimensional manifold. The interior is a noncompact n-manifold. The reader should note that the terms “interior” and “boundary” were used in the preceding paragraphs in a sense different from that which is usual in point set topology. However, this will seldom lead to any confusion. Examples show that a manifold with boundary may be compact or noncompact, connected or not connected.

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