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: