5

CHAPTER I

THE PROBLEM, METHODS OP ATTACK, DIFFICULTIES

1. The representation problem for

Fre*ohet varieties

1.1 The representation problem is easy to state in spite of the fact

that it appears to be somewhat difficult to solve.

The basic notion is that of a mapping f\X - Y; that is, a

continuous transformation from one space into another. The notation and

terminology of transformations to be employed here will follow Lefschetz

[Q, pp. 2 - 3]; unless otherwise stated, however, the spaces X and Y will

be metric, and for brevity a statement such as "f is a mapping" will some-

times be employed without explicitly naming X or Y.m In addition to these

common conventions, a mapping fiX — Y is said to be trivial [non-trivial] if

f(X) is degenerate; that is, a single point [non-degenerate; that is, consists

of more than one point]. Moreover, f:X z* Y means that / is from X onto

Y; that is, f(X) = Y, while h:X % Y means that h is a homeomorphism

from X onto f, (The notation X £ Y is read J is homeomorphic to Y).

It should be mentioned that the symbols - * and zx are much too

useful to be reserved exclusively for the purpose indicated above. If X is

a metric spaceand *n € X, n = 0, 1, 2, • •-, thenthenotation *n - xQ means

x converges to xQ; that is, if p is the distance function in X, then

lim pixn, xQ} = 0. If fn:X - Y is a mapping, where X and Y are

metric, n = 0, 1, 2, • • •, then the notation fn - » /0 means that /n con-

verges to f0; that is, x £ X implies /n(*) " * /(*)• The notation

/n z:/0 means that fn converges uniformly to f0; that is, if pifn /} =

sup pifn(x), f0(x)}, x £ X, where p is the distance function in Y, then

1.2 With this background, suppose I is the class of Peanospaces(that

is, the totality of spaces each of which is the image, under some map-