i . e ]
THE PROBLEM
9
each o f which i s homeomorphic t o th e range space o f any representatio n of F.
Moreover, i t i s now p o s s i b l e to c a t a l o g Frechet v a r i e t i e s , F, i n terms o f
the associate d c l a s s e s , [F] . For example, a Frechet variety , F, i s c a l l e d
a Frechet curve i f and only i f [F] i s the c l a s s of 1 - c e l l s ( = close d arcs)
or 1-spheres ( = Jordan curves). Indeed, subolassifioations are possible, for
If [F] i s the class of 1-cells, then the Frechet curve, F, i s known as a
Frechet curve of the type of a 1-cell; i f [F] i s the class of 1-spheres,
then the Frechet curve, F, i s known as a Frechet curve of the type of a
1—sphere.
A Frechet variety, Y i s called a Frechet surface if and only i f
[F] i s in the class of 2-manifolds ( = compact connected 2-manifolds with or
without boundary). Consequently, in the spirit of further subclassification,
i f [F] i s the class of 2-spheres, the Frechet surface, F, i s known as a
Frechet surface of the type of a 2-sphere; i f [F] i s the class of 2-cells,
the Frechet surf ace is known as a Frechet surface of the type of a 2-cell; etc.
In general, a Frechet variety, F, i s called a Frechet n-manifold
if and only if some space in [F] i s an n-manifold. (This classification i s
certainly not exhaustive, as there are Peano spaces which are not manifolds.)
1.8 So much for what can be done with the first necessary condition. A
further property results as a consequence of the second necessary condition,
namely, that fitfj) = /^(^)» This shows that the image of the range space
i s independent of the representation. Hence with each Frechet variety, Y,
there i s associated a particular Peano space, F*, which i s precisely f(I)
for any representation, f:X -* 7, of F. This Peano space, F*, associated
with the Frechet variety, Y, i s not to be confused with the variety itself—
F* i s a Peano space, while Y i s an equivalence class of mappings. In fact,
a source of some misunderstanding i s the fact that i f one is given two Peano
spaces, I and 7, the former being non-degenerate, then there i s a Frechet
variety, Y, such that [F] i s the class of spaces homeomorphic to J, and
F* = 7* In other words, there i s a mapping f:I 3 7 (see Whyburn [10, p. 34,
Theorem 4.6]). Consequently, i t i s possible, to take a simple example, for Y
to be a Frechet curve and for F* to be a 2-sphere - a situation which might
tempt one to call Y a surface. This shows that knowing the space F* i s of
absolutely no help in classifying the Frechet variety Y as to type; i t i s
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