From: Lee Corbin (lcorbin@tsoft.com)
Date: Tue Aug 20 2002 - 19:53:20 MDT
Rafal writes
>> Anders Sandberg wrote:
>>
>> On the other hand, you could have a closed universe where the volume is
>> infinite. Think about the globe analogy and imagine a earth map wrapped
>> twice around the globe... One could imagine this wrapping continuing
>> even further and in all directions
>
> ### I am not sure you could call this a closed universe - after all, this
> would be the result of connecting of an infinite number of units (each equal
> to an Earth surface) but with different contents. Even if the local topology
> was that of a sphere with a finite radius, the structure as a whole would be
> infinite, that is, its inhabitants would have infinite lines of sight, which
> would be the opposite of closed.
What meaning of *closed* is this, now? Not having "infinite lines of sight"?
Maybe you mean by "closed" straight lines (geodesics) of infinite length.
(For example, in a 2 dimensional square where the top is identified with
the bottom, and the left side with the right side, one has two kinds of
straight lines: those that form a quick loop, i.e., start from the middle
of the left side and go straight across to the middle of the right side
and at that point intersect themselves, and those others that are at an
irrational angle, and so are truly infinite in the other meaning.)
So there seem to me to be three meanings of closed possibly
meeting here: (1) topologically closed, i.e. a set containing
all its limit points and, if it has one, containing its boundary
(2) closed as having infinite non-intersecting geodesics, and
(3) closed as in universes that eventually collapse.
For each meaning, one should always indicate whether *closed*
is the opposite of *open*. In topology, for example, the
terms are not mutually exclusive.
> For a closed n-dimensional universe with an infinite volume I could suggest
> an n-dimensional surface folding as a fractal in an n+1 space. I don't have
> the oomph to think this through.
This one would be a good one for us to examine. I'm not
sure I've followed or understood everything in this thread.
(These days, I'm not at all sure I'm even receiving all the
outgoing posts!)
Your closed n-dimensional universe with infinite volume
would support infinitely long geodesics, no? Now I'm
again doubtful that I even understand what is meant by
"closed infinite volume" universe; I can't think of an
example.
Lee
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