[FRIAM] Manifold Clarification

[Jon Zingale]

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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