Re: Complement of zero dimensional space

On Jun 28, 4:20 pm, Denis Feldmann <feldmann.denis.asuppri...@xxxxxxx>
wrote:
victor_meldrew_...@xxxxxxxxxxx a écrit :

On 28 June, 18:01, "Dim BandTech.con" <tttppp...@xxxxxxxxx> wrote:

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.

Can you define the three sets of dimension zero needed to
cover the complex plane (your P_3) using your polysigns?

Or are you just lying again?

Lying is not the right word. Spam would be more like it

Every question that anyone asks can be contextually corrected within
the answer. Dimension is a discrete behavior. for instance we do not
generally observe 2.12 dimensional spaces do we? The ray as a simpler
form than the line is a very simplistic perspective regardless of
polysign. The fact remains that three rays will represent 2D space
accurately. In that the real line as one dimensional has two rays,
then beneath lays a single ray space and while it is not immediately
apparent from traditional math, extending the behaviors of the two ray
and three ray systems downward onto that single ray do imply that it
is zero dimensional. This is nearly a description without polysign,
yet in hindsight we have just built the fundamentals for generalizing
sign from the perspective of geometry. Nicely the unification of space
and time is possible due to the graphical nature of the unidirectional
ray as time. Yes, three of these rays do form the fundament of the
plane, but their joinery must be enforced via
( 1, 1, 1 ) = 0
or more generally
( x, x, x ) = 0
or in polysign
- x + x * x = 0
which states that the three rays are in balance and thus do form the
plane as a vector space. Again, stepping back to the two-signed
numbers (the reals) we see the traditional definition of dimension
built upon two rays. The idea that two components might compose the
real number is a point of contention for some, however the usual
representation can always be gotten to since the two rays cancel each
other so that for instance
- 3 + 5 = ( - 3 + 3 ) + 2 = 0 + 2 = + 2 .
The same is true of each dimension. Whether there is some utility to
the additional components is something I revisit occasionally, but
there is no inherent conflict with either interpretation, much as an
integer might be cast into a real without contention.

Anyway in terms of the title of this thread I do not know how to
complement the one-signed numbers. In the context of three-signed
numbers which are 2D and somehow have become a focus there is an
argument that taking away one ray then yields the real numbers, but
there is likewise support for a claim that this construction
represents one third of the plane. I don't care for either of these
constructions. Any requirement of a union as in set theory here does
not seem appropriate.

Whether cartesian 'products' are really superpositions or products is
also a thorny issue from my perspective. Neither of these words truly
embodies the cartesian product since by definition there is no
operation between those independent components. This independence is
troubling, for upon declaring independence what right has one to
enforce any relation between them? Claiming a functional relationship
breaks the symmetry badly. I'm not about to argue that all of
mathematics which relies upon cartesian products is broken, but I do
feel that since an alternative exists that such considerations are
only natural to run through in order to assess the situation.

- Tim
.

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