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-
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