Abstract
This chapter presents a certain open manifold whose group is unity. Any infinite, unbounded manifold, in which every finite circuit bounds a 2-cell and every finite 2-cycle bounds a finite region, is what one shall call a formal 3-cell. By a formal 3-cell is meant an infinite, unbounded manifold, a subdivision of which, say C , contains a n infinite sequence of elements E1 E2,…, such that En+1 contains every solid of C that meets En. Under these conditions, it is obvious that any solid and any finite region in C is contained in En for some value of n. It is not difficult to show that a subdivision of C has a rectilinear model covering Euclidean 3-space; the symbol for such a rectilinear complex is a formal 3-cell. Thus, an infinite manifold is a formal 3-cell if and only if its rectilinear model in Hilbert space is in (1, 1) semilinear correspondence with Euclidean 3-space.
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