Properties of Sweep

[Ross A. Finlayson]

Sweep: "the field obtained by the ultrapower construction
from the space of all real sequences", -- "Hyperreal number: Properties"
-- https://en.wikipedia.org/wiki/Hyperreal_number#Properties

Also countable and a metric space. (Domain and range.)

Here for example both domain and range of function sweep
are domains.

Domain and co-domain, range and domain,
the natural, 0, 1, 2, 3, ..., unit, [0,1] or zero to one,
is giving the integer domain a denominator
and finding it the integer range,
numerator and denominator.

lim d->oo n->d n/d

Correct in the middle there...
right in the middle.

lim denominator->oo numerator->denominator numerator/denominator

Conveniently goes with saying,
naming a usual prototype of an integer value metric space,
here for example it's often mostly neatly written t->oo,
t for time.

Then time related problems are always well-defined.

[Ross A. Finlayson]

Line drawing results in a real ultrafilter,
also for example space filling.

The line drawing counterexample to the
powerset uncountability result of the
points of the line, line drawing or sweep
this function, is countable, but, a continuous
domain in the language of 2^w is uncountable,
here that the line drawing is in the space besides
the language. Here the line drawing is "all through
the language". Because the integers are countable
and exhaust, the domain is countable but complete.

Two resulting _different_ "sets" in the least properties
they need in their theories, these points or regions
in the integer plane, still they share for example
all the integers and paths in between them.

I.e., you can't make a line with points except drawing it,
or that the sweep fills the area of the space,
or it results in the language of the uncountable language,
instead of the countable construction of the set of the
ratios of the integer points.