1.7] THE PROBLEM 11 -• Y X The factorization is not, strictly speaking, unique — more will be said of this later. (Double arrows will be omitted from diagrams in the future.) No w suppose that f±: X ± -* Y an d f Q : %Q ~* Y a r e Frechet equiva- lent, while Ij*! a n d ^2mQ a r e naonotone-light factorizations of f± and fq with middle spaces X1 and X^, respectively. Having successfully completed a similar body of research in case J- and T are 1-spheres, Kerekj arto' sought to show that there was an exact matching of the light fac- tors of the two mappings. In this endeavor Kerekj arto' was successful specifically, he was able to show that there is a homeomorphism h:Xx £ X^ such that I = IJi. To be quite precise, Kerekjarto showed this to be the case if X1 and X^ are both 1-spheres or both 2-spheres. In fact, this i s true in ultimate generality within the class of Peano spaces, as was indioated at the beginning of 1.7. One can now prove the following statement. If fx'Xx -* Y and / • J _ - Y are two light mappings which are Frechet equivalent, then there is a homeomorphism h:X± ~ I such that fx = f^h. The proof uses the fact that the identity mapping k^:X^ z* J . (defined by the formula ^{(x{) = x { *{ £ Xj) is monotone, i - l, 3. Hence f{^i Is a monotone-light factorization of / j with middle space X±, i = l, a. The result follows on employing the Kerekj arto' necessary con- dition. The Kerekj arto' condition was the first non-trivial necessary con- dition for Prechet equivalence and may be re-stated in a highly convenient form. If fx ~ fQ, then there are monotone-light factorizations, / = lmx and f2 ~ I™*' w i - t n ^ n e s a m e middle space. (The mapping I here is I^ above, while the mapping m± is hm±.) In other words, if for every e 0 there is a homeomorphism h£ such that:

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