Topology of Numbers
pi.math.cornell.edu
pi.math.cornell.edu
"By special arrangement with the publisher, an online version will continue to be available for free download here, subject to the terms in the copyright notice."
Some details about how he typesets his books here: http://pi.math.cornell.edu/~hatcher/AT/typography.html
It’s interesting to see the range of applications this structure has. While it helped me understand a few ideas like how a Cauchy sequence might work, practical applications included things like finding approximate ratios for floating points with various limits on the scale of the denominator and single update list sorting systems that don’t rely on midpoints (they run out of precision and require relabeling rather early). The history of the concept is also worth looking at.
A good overview is available in Graham, Knuth, and Patashnik‘s book “Concrete Mathematics.”
> we are using the word "Topology" in the general sense of "geometrical arrangement" rather than its usual mathematical meaning
> perhaps the title could have been "Topography of Numbers" instead.
https://www.google.com/search?q=define+topology
> 2. the way in which constituent parts are interrelated or arranged.
> "the topology of a computer network"
Every set is open, so every set is closed. So something like a closed interval, or the product of closed intervals, is just going to look like the network that defines a line segment, square, etc... right?
In that sense, it seems like the geometry of numbers and the topology of numbers are basically the same thing.
But I'm going to be honest -- I know basically nothing about arithmetic geometry.
Instead you can look at things like prime ideals of integers localized at some prime, and consider algebro-geometric topologies on that
I think the language and machinery of topology, even when just reconstructing the language of sets in a discrete setting, highlights interesting facets of numbers.
eg, if you look at the inverse image of various mappings, and particularly in cases where you can iterate this via a function from a set into itself, you can start building up meaningful comments on certain classes of number theory problems.
But I am curious what you mean by "prime ideals of integers localized at some prime", since I know what (prime) ideals are, but am not sure I follow what you mean by localized
Look up ring localization.
They're obviously equivalent.
My point is that what's easily noticeable in one incarnation of the theory is different than what's easily noticeable in the other incarnation (or if you prefer, expressible), and switching our language for the same abstract structure can highlight different interesting features of it. And further, there's still utility to using topological perspectives and language to discuss the integers or naturals, even if it's equivalent to set theory.
I do appreciate the reference to ring localization -- will have to look at that further.
Just to address your original comment in this thread, perhaps it's relevant to note the following. Consider the homotopy theory of the category of nice topological spaces. The full subcategory of topological spaces supported on discrete topological spaces inherits a homotopy theory. This inherited homotopy theory is equivalent to the trivial homotopy theory on the category of sets: where weak equivalences are isomorphisms. This is the sense in which discrete spaces don't have an interesting homotopy theory, at least naively.
(This statement you can precise in your favorite model for the homotopy theory of spaces, via infinity categories, model categories etc.)
It's stunning how intuitive geometric explanations can feel. I like the geometric approach.