Re: Standard Topology on S^n ?


On 27/2/08 12:25 PM, in article 7ge9s35e9gii82tdpiq8jqhfqgpofpes99@xxxxxxx,
"Gonçalo Rodrigues" <nospam@xxxxxxxxxxxx> wrote:


Don't you think that these R^n things are extraordinary contortions,
logically ? Why should spheres depend on R^n for their topological
authenticity ?


Don't understand your questions. What exactly is the extraordinary
contortion about R^n, or why should a sphere, or any topological
space, be less "authentic" (whatever that means) just because it
arises as a subspace of a larger one?

My spheres arise as follows. I go outside. I standstill in R^3= at a
"fixed" point. My eyes traverse a topological hemi-sphere known as the Sky
= the observable universe by moving through angles without moving through
displacements. I postulate a conjugate observer to form a complete sphere
which I think you call S^3. Although there is some confusion in mathematics
on the numeric. R^3 is irrelevant to S^3; at least to me.

I do understand that mathematicians play a different game. They choose to
assume they can ignore physical reality and take up any arbitrary abstract
approach they choose - constrained only by a kind of logic. But any game,
to an outsider appears ridiculous. No doubt my games look ridiculous to
you.

That after all is the (mathematicians' - my insertion TP) definition
of (the unital) sphere -- the set of all points in R^{n + 1} that are
at the same distance 1 from the center...

Which I think is contorted, peculiar, queer. irrational, unrealistic,
unjustified, wrongly motivated, ......

After all topologically a sphere is most unlikely to be unital, except in
very abstract arbitrary situations. In fact topologically a sphere need not
ever be spherical ! So starting from a definition that imposes unital
sphericity is asking for trouble, IHMO.

I hesitate to go into mathematical deep water but are there not non-metric
definitions of R ?

Regards,
G. Rodrigues

Thanks for your patience with my musings. No doubt you have more important
things to attend to.

.

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