I was hoping you would sketch out more of your objection to my claim that the Alexander horned sphere provides an example of a fractal space whose topology is given simply as a sphere. In speech, I can feel pressured to make the best of the few words I have room to express and sometimes at the expense of accuracy. Analysis of the Alexander horned sphere (and the space it encloses) is a bit more nuanced than I let on, the details of which may be helpful for our discussion of clouds. OTOH, Friam discussions are sometimes nothing short of a bombastic free-for-all where injecting aporia or the occasional first order footnote is about *as good as one can hope for*. If it turns out to help our discussion here, I will dust off my copy of *Hocking & Young*.
Yes, a discussion of limit points would be necessary for investigating the topology of this pathological object. Analysis of its interior and exterior yield very different results, while the ball is simply connected its boundary is not. Somehow, this off-the-top-of-my-head example seemed to be relevant enough to Nick's itch that I hoped it would slow things down. Nick, Steve, Frank, et al. Before we dive into Mandelbrot thumping, or some other obnoxious witch hunt of popular mathematics, what exactly is our goal? I don't mind beginning with Nick's definition of a cloud, but only if that means we work to prove what is and what is not an entailing theorem. Further, I will hope that we *do not* confuse these theorems for truths about our material world. I maintain that any definition we start with will have *some* domain of applicability, but we are far too along in our understanding of rhetoric (as a culture) to waste time building strawmen. Granted this, if Nick wants to use *shrouds* as a way of talking about Darboux sums <https://en.wikipedia.org/wiki/Darboux_integral> converging to Riemann Integrals, say, well fine. I am not entirely sure there is any particular reason we need to dive into an analytic hole, but hey. Nick, if there is a question underlying all of this demand for technology, please state it. EricC, however, helped me to feel justified in claiming that asking *what is a cloud, really* is not a productive question. Jon
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