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 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 is hm±.) In other words, if for every e 0
there is a homeomorphism such that:
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