A Translation of the Normal Moore Space Conjecture
R. H. Bing · Proceedings of the American Mathematical Society · 1965
The purpose of this note is to translate an unsolved problem in topology into a nontopological setting so that it can be considered by a wider audience.Those interested in the foundations of set theory might find the new formulation more to their liking.A quickening of interest in logic generated by Paul Cohen's result that the Continuum Hypothesis is independent of the axioms of set theory (Zermelo-Fraenkel or Godel-Bernays) together with the Axiom of Choice suggests that this is an opportune time to consider this reformulation.Let X denote a set of points, R the cartesian product XXX, and L the diagonal of R consisting of all points (x, x)EXXX.It may be convenient to think of R as a unit square in the Euclidean plane with diagonal L from (0, 0) to (1, 1) as shown in Figure 1.However, we do not insist that the cardinality of X be that of the continuum.(0,0)The horizontal projection h(x, y) = (y, y) and the vertical projection v(x, y) = (x, x) sends R onto L. We shall be concerned with sets W as shown in Figure 1 such that h(W)C\v(W) = 0.Although W may be as shown, it may be more complicated (such as the set of points