Closure
If you take a number line
or rather the number line
the real number line with all the
infinitely infinite decimal expansions
accordioned up into a single string segment
that you can hold in your hand.
If you take this piece of string
wrap the ends around your finger
tying them together at the ends to create a loop
you will then have a model for the projective closure of ℝ.
And odds are good you don't particularly know what that means.
Pinch the knot in your fingers
adjust it to sit at the peak of your knuckle
because this knot represents infinity
rotate it clockwise one full turn, leaving it in exactly the same place
but it represents negative infinity now
but it always did
the two hold the same position in space
This time hold your palm supine
search for the midpoint of the string
nestled into a seam on your finger,
look for it now. try to find zero.
it exists as a single perfect point, you know it does, it has to,
but it isn't flashy like the infinities
you might have to trust in the precision of nothing
in the infinitesimal fulcrum of your joints
And with our poles defined, the reals
start pouring like honey out from the origin
saturating the number line with all possible strings of all possible digits
decimal expansions sticking together, inseparable for all finite purposes,
for all concrete purposes
Except of course that we all use numbers daily
and look you're holding them all, right there in your hand
its just the way they bleed together if you hold them all at once
and atomicity is lost