Find topologies that characterise smoothness

Asked Aug 27 '23 at 15:03

Active Aug 31 '23 at 11:18

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Is there a family of topologies $\tau_i$ on $\mathbb R$ such that for every map $f\colon \mathbb R \to \mathbb R$ we have $$ f \text{ is continuous w.r.t. }\tau_i \forall i \iff f \text{ is smooth?} $$

By smooth we mean infinitely often differentiable here.

Related is this article, proving that no topologies on $\mathbb R$ can characterize differentiability as continuity. However maybe if we consider a whole family of topologies?

edited Aug 31 '23 at 11:18

asked Aug 27 '23 at 15:03

Alex

5

Since the meaning of "smooth" varies in the literature (continuous derivative? infinitely many derivatives? this notion? etc.), you should include a definition of what you mean by "smooth". That said, somewhat related are: Topologically, is there a definition of differentiability that is dependent on the underlying topology, similar to continuity? (continued) – Dave L. Renfro Aug 27 '23 at 18:14
AND Which functions of one variable are derivatives ? AND the 1976 paper Derivatives: why they elude classification by Andrew M. Bruckner AND the 1995-1996 survey paper The problem of characterizing derivatives revisited by Andrew M. Bruckner (continued) – Dave L. Renfro Aug 27 '23 at 18:14
AND the 1997-1998 paper Characterizing derivatives by preimages of sets by Krzysztof Ciesielski. – Dave L. Renfro Aug 27 '23 at 18:14

Find topologies that characterise smoothness

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