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