Topologizing different classes of real functions


Krzysztof Ciesielski

Canad. J. Math. 46 (1994), 1188-1207

The purpose of this paper is to examine which classes F of functions from Rn into Rm can be topologized in a sense that there exist topologies \tau1 and \tau2 on Rn and Rm, respectively, such that F is equal to the class C(\tau1,\tau2) of all continuous functions f:(Rn,\tau1)-->(Rm,\tau2). We will show that the Generalized Continuum Hypothesis GCH implies the positive answer for this question for a large number of classes of functions F for which the sets {x: f(x)=g(x)} are small in some sense for all different f and g in F. The topologies will be Hausdorff and connected. It will be also shown that in some model of set theory ZFC with GCH these topologies could be completely regular and Baire. One of the corollaries of this theorem is that GCH implies the existence of a connected Hausdorff topology T on R such that the class Lin of all linear functions g(x)=ax+b coincides with C(T,T). This gives an affirmative answer to a question of Sam Nadler. The above corollary remains true for the class P of all polynomials, the class A of all analytic functions and the class of all harmonic functions.

We will also prove that several other classes of real functions cannot be topologized. This includes the classes of Cinfinity functions, differentiable functions, Darboux functions, and derivatives.

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Last modified April 24, 1999.