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Topologizing Different Classes of Real Functions

Published online by Cambridge University Press:  20 November 2018

Krzysztof Ciesielski
Krzysztof Ciesielski*
Affiliation:
Department of Mathematics, West Virginia University Morgantown, West Virginia 26506-6310, U.S.A.
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Abstract

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The purpose of this paper is to examine which classes of functions from can be topologized in a sense that there exist topologies τ1 and τ2 on and respectively, such that is equal to the class C(τ1 , τ2) of all continuous functions . We will show that the Generalized Continuum Hypothesis GCH implies the positive answer for this question for a large number of classes of functions for which the sets {x : f(x) = g(x)} are small in some sense for all f, g ∈ f ≠ g. 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 such that the class L of all linear functions g(x) = ax + b coincides with . This gives an affirmative answer to a question of Sam Nadler. The above corollary remains true for the class of all polynomials, the class 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 C∞ functions, differentiable functions, Darboux functions and derivatives.

Keywords

26A1554H1003E35topologies on Rnlinearanalytic and harmonic functionsderivatives
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 46 , Issue 6 , 01 December 1994 , pp. 1188 - 1207
Copyright
Copyright © Canadian Mathematical Society 1994

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