Functions characterized by images of sets


Krzysztof Ciesielski & Dikran Dikranjan & Stephen Watson

Colloq. Math. 77 (1998), 211-232.

For non-empty topological spaces X and Y and arbitrary families A subset of P(X) and B subset of P(Y) we put CA,B ={f:X-->Y| f[A] is in B for every A in A)}. In this paper we will examine which classes of functions F subset of Y^X can be represented as CA,B. We will be mainly interested in the case when F=C(X,Y) is the class of all continuous functions from X into Y. We prove that for non-discrete Tychonoff space X the class F=C(X,R) is not equal to CA,B for any A subset of P(X) and B subset of P(R). Thus, C(X,R) cannot be characterized by images of sets. We also show that none of the following classes of real functions can be represented as CA,B: upper (lower) semicontinuous functions, derivatives, approximately continuous functions, Baire class 1 functions, Borel functions, and measurable functions.

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Last modified September 16, 1998.