[FRIAM] Manifold Enthusiasts

[Steven A Smith]

Frank -
> Perhaps annoying clarification.  The word closure has two important
> meanings.
>
> In a topological space a set is closed if it is the complement of an
> open set.
>
> In the more concrete case of Rn, which we're talking about, a set is
> closed if it contains it's limit points (boundary).
>
> The other use of closure is the one Glen(?) mentioned.  The sum of two
> elements in the set is in the set and the same for other operations.
>
> Barry, help!

Yet more good elaboration/correction.  I think for Nick's benefit, the
second one is *most* relevant, comes closest to your own original
common-sense idea of "walking around on a surface without falling off"? 
Cannot 2 be derived from 3 with the right assumptions?

Also, I hope your nerve pain is subsiding...  I guess this bodes poorly
for your tennis game?  At least overhead slams and solid serves, unless
it is your off-hand that is afflicted?

I'm unfamiliar with the pain meds you mentioned (really all outside of
the basic salisylic acid, acetomenophen, and ibuprofen.   I was hoping
your report of the two med's efficacy would lead to some elaboration on
the question of change of behaviour without change of brain state?  
Where do these meds act?  In the brain tissue itself?  On the pinched
nerve?  Somewhere in between?   And does that just beg the question over
into whether the spinal chord and the entire nervous system are
nominally "part of the brain"?

- Steve