8
THE REPRESENTATION PROBLEM
[I
(0
i f 0  x £ i/Q.
Now for any
/*(* )
/*(* )
€
0,
he(x)
iQX  1
=
X,
l e t
=
' 2€X
s ( i

i f ±sz
i f 0 £ x
l .
* ' * .
[Q(I
 e ) x  ( l  3€)
Then:
P ^ W , / ^
e
( x ) }
3 6 X
i f i / s £ x £ i,
i f 0
IseC*  x) i f *•« £ x £ i .
Therefore:
P { / ^ / ^
€
e, and ^ ~ / ^
On the other hand, i t i s quite obvious that any attempt to obtain
a homeomorphism h:X% £ X^ such that f
x
= fQh i s foredoomed to failure
because f± maps the segment 0 ^ x £ i / s onto a single point, while /^/t
i s topological.
This simple example shows that approximate matching is an essen
tial feature of Frechet equivalence, and the obvious attack on the problem
f a i l s . This being the case, i t i s well to consider a few simple necessary
conditions which result as a consequence of the hypothesis that / ~ /
in the hope that a large body of such necessary conditions wil l supply a
sufficient condition.
1.5 A first necessary condition is that the range spaces Xx and X2
he homeomo rphi c
A second necessary condition is that /^C^)
=
/gC^)*
Both conditions are immediate consequences of the definition of
Frechet equivalence (see 1.2), and show, by the way, two highly important
properties of Frechet varieties . As these properties are often misunder
stood, a few comments will not be out of place.
First, i f Y i s a Frechet variety with representations f±:^± " * ^
and f9:*9 "* Y, then the fact that X± and X9 are topologically equivalent
makes i t possible to associate, with V, a class, [F], of Peano spaces, Y,