Chapter 1 A Little Set Theory / see it, but I don't believe it. CANTOR TO DEDEKIND 29 JUNE 1 When we say that space has three dimensions, what do we mean? HUREWICZ AND WALLM Functions are the single most important idea pervading mod mathematics. We will assume the informal definition of a functio a well-defined rule assigning to each element of the set A a un element in the set B. We denote these data by / : A B the rule by / : a G 4 H f(a) G B. The set A is the domain / and the receiving set B is its codomain (or range). We make important distinction between the range and the image of a funct f(A) = {/(a) G B | a G A}, which is a subset contained in B. When the codomain of one function and the domain of ano coincide, we can compose them, that is, given f: A -^ B, g: B we can define g o f: A C by the rule g o f(a) = g(f(a)). X C A, then we write f\x '• X B for the restriction of the of / to the elements of X. This changes the domain and so a different function. Another way to express f\x is to define http://dx.doi.org/10.1090/stml/031/01
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