AN ELEMENTARY ROMP THROUGH SOME ADVANCED EUCLIDEAN TOPOLOGY

In a discussion following R. H. Bing's talk on "The elusive fixed point property " [2] at the 1968 Houston Topology Conference it was observed that the easy proof of Brouwer's Fixed Point Theorem for an interval, obtained by considering points which move right and those moving left, extends to two dimensions with the aid of a Phragmen-Brouwer property but no further. The author tried for some time to find the proper extension of this property with little success. Upon reading a new solution of this two-dimensional problem (and others) by Albrecht Dold [3], the proper extension was eventually found to be a variation of what is sometimes called Alexander's Addi tion Theorem. Since it is really a restatement of the latter, it is not a new result but is a new way of interpreting and relating known results. This new approach permits very short simple proofs, usin