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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-
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