fom
Posts:
1,969
Registered:
12/4/12


Re: Prime Interger Topology
Posted:
Oct 20, 2013 10:42 PM


On 10/20/2013 9:17 PM, William Elliot wrote: > On Sun, 20 Oct 2013, fom wrote: > >> By the way, I just formulated an arithmetic based on >> my set theoretic axioms and the fact that the natural >> numbers enjoy a topology called the prime integer topology. > > The prime integer topology for N is the topology generated > by the subbase Bs = { (n + pN) /\ N  n in Z, p prime } > > What if the prime integer topology were applied to Z? >
I am not really applying it. I am constructing it. Or, better, I am trying to construct as much of it as needed.
The integers loosely relate to the naturals via absolute values. So, here is part of the expository for my difference relations. Note that the difference relations actually introduce the notion of a directed set by virtue of the "commutativity" of what needs to be expressed,
In fact, it is the difference relation that introduces a usable order...
<beginquote>
The axiom for difference relations is given by,
AwAxAyAz( (w,x,y,z) <> ( ( +(w,x,z,y) /\ (w,y,x,z) ) /\ ( ~( (w,y,z,x) ) /\ ( ~( (w,x,z,y) ) /\ ( ~( (w,z,x,y) ) /\ ~( (w,z,y,x) ) ) ) ) ) )
The first conjunct in the expression,
+(w,x,z,y) /\ (w,y,x,z)
asserts that there must exist an additive relation by which z is the sum of x and y. The second conjunct asserts that in relation to such a sum, the addends of the sum are commutative with respect to difference relations.
This expresses,
8  3 = 5
8  5 = 3
in relation to
3 + 5 = 8
Observe, by this requirement in the definition of the related relations, that the directed set structure is imposed on the domain.
Difference relations are not simply "the opposite" of additive relations.
The expression
( ~( (w,y,z,x) ) /\ ( ~( (w,x,z,y) ) /\ ( ~( (w,z,x,y) ) /\ ~( (w,z,y,x) ) ) ) )
shows the relationship of the difference relations to order on the domain. Since difference relations are understood with respect to sums, the sum of two terms may not occur in specific indices of the relation.
That is, the fourth term in a difference relation is always greater than the second and third terms.
<endquote>

