Re: Complement of zero dimensional space

On Jun 28, 12:12 am, Denis Feldmann <denis.feldmann.sanss...@xxxxxxx>
wrote:
G. A. Edgar a crit :
In article <20090626221727.U8...@xxxxxxxxxxxxxxx>, William Elliot
<ma...@xxxxxxxxxxxxxxxx> wrote:
On Fri, 26 Jun 2009, G. A. Edgar wrote:
<ma...@xxxxxxxxxxxxxxxx> wrote:
Conjecture. If S is a zero dimensional subspace of R^2,
then R^2 - S is a dense, path connected subspace.
A topological space has topological dimension n if and only if it can
be written as the union of n+1 sets of dimension zero.
A = [0,1] /\ Q and B = [1,2] /\ Q are two zero dimensional spaces.
Thus C = A \/ B = [1,2] /\ Q is one dimensional?
Hmmm... I was a bit off... How's this:
A topological space has topological dimension at most n if and only if
it can be written as the union of n+1 sets of dimension at most zero.
or
A topological space has topological dimension n if and only if it can
be written as the union of n+1 sets of dimension zero and cannot be
written as the union of n sets of dimension zero.

This is ridiculous. Which are the 3 sets (of dimension 0) whose union
is R^2 ? Why dont you use *boundaries* ? (seehttp://en.wikipedia.org/wiki/Inductive_dimension)

Polysign numbers do answer this question directly, though the term
'union' is not quite the proper construction.

Since traditionally dimension is expressed in terms of the real number
which is actually bidirectional then the answer simply lays there. One
dimension means two directions. These directions happen to be opposed
exactly such that
- 1 + 1 = 0 .
To generalize this directional logic rather than step up to RxR we
might choose to consider a three directional construction
- 1 + 1 * 1 = 0
where '*' is a new sign and the result is a new class, not a union
from the old two-signed variety. These turn out to be the complex
numbers when superposition and product are fully defined:
http://bandtech.com/polysigned
Beneath the bidirectional reals lay a unidirectional construction
which is required to be zero dimensional by extension of the
expressions above:
- 1 = 0 .
While paradoxical at first this expression and the unidirectional
concept form perfect time correspondence. These cancellations
expressed here are not actually necessary arithmetic in terms of the
algebra that can be done on these number systems. For instance
- 1 - 3 = - 4
( - 2 )( - 3 ) = - 6
are valid operations on one-signed numbers. The cancellation law is
performed directly when rendering any of these number systems
graphically. This is a new concept along with the usage of the ray as
a fundamental constructor rather than the line. Spacetime
correspondence follows quickly within this generalization and so the
potential depth of breaking dimension down to sign is not only
coherent, but it is practical.

Dimension is sign, just off by one. There does seem to be a caveat
which has not been covered very well to date. On a 2D medium such as
piece of paper we are free to project up a 1D line or for that matter
a 0D ray. Yet as well a point may be drawn on the paper and so the
graphical representation may still have some resolution which has not
yet been offered. Does magnitude inherently contain its lessers? This
question might at first seem tangential but I believe it is nearby to
the awareness that I am proposing. Such slender interpretations are
important. We are so accustomed to the usage of the 2D format for our
work. Could this be part of the problem? We are now back to scrolling
media here on these machines... wouldn't it be something if there is a
rabbit to be pulled out of the scrolled ***? There is something
fundamental which remains to be constructed.

- Tim
.

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