6 ms·

What a confusing name in an era where homotopy-theoretic methods in arithmetic geometry are flourishing. Looks Hatcher covers non of that and use "topology" to

What a confusing name in an era where homotopy-theoretic methods in arithmetic geometry are flourishing. Looks Hatcher covers non of that and use "topology" to mean "geometrically motivated".

gnulinux 8y ago
I agree, I can't stop scratching my head. Maybe "Geometry of Numbers" is a better name.
- yesenadam 8y ago
  Geography maybe?
gowld 8y ago
http://pi.math.cornell.edu/~hatcher/TN/TNpage.html http://pi.math.cornell.edu/~hatcher/TN/TNpage.html > 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 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"
NotAnEconomist 8y ago
This is a dumb question, but what's the difference between geometry on discrete sets and homotopy theory on discrete sets, eg, in the spirit of digital topology? 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.
- dbranes 8y ago
  As you observe one can't really recover any non-trivial number theoretic things by looking at integers with the discrete topology. The theory of discrete topological spaces is just the theory of sets. Instead you can look at things like prime ideals of integers localized at some prime, and consider algebro-geometric topologies on that
  - NotAnEconomist 8y ago
    Well, hold on now. 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
    - dbranes 8y ago
      The category of sets embeds into the category of topological spaces as a full subcategory, the essential image of which are the discrete spaces. Hence the equivalence I claimed is a precise statement. Look up ring localization.
      - dbranes 8y ago
        (the embedding is full & faithful)
      - NotAnEconomist 8y ago
        > Hence the equivalence I claimed is a precise statement 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.

What a confusing name in an era where homotopy-theoretic methods in arithmetic geometry are flourishing. Looks Hatcher covers non of that and use "topology" to mean "geometrically motivated".

gnulinux 8y ago

I agree, I can't stop scratching my head. Maybe "Geometry of Numbers" is a better name.

yesenadam 8y ago
Geography maybe?

gowld 8y ago

http://pi.math.cornell.edu/~hatcher/TN/TNpage.html http://pi.math.cornell.edu/~hatcher/TN/TNpage.html > 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 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"

NotAnEconomist 8y ago

This is a dumb question, but what's the difference between geometry on discrete sets and homotopy theory on discrete sets, eg, in the spirit of digital topology? 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.

dbranes 8y ago
As you observe one can't really recover any non-trivial number theoretic things by looking at integers with the discrete topology. The theory of discrete topological spaces is just the theory of sets. Instead you can look at things like prime ideals of integers localized at some prime, and consider algebro-geometric topologies on that
- NotAnEconomist 8y ago
  Well, hold on now. 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
  - dbranes 8y ago
    The category of sets embeds into the category of topological spaces as a full subcategory, the essential image of which are the discrete spaces. Hence the equivalence I claimed is a precise statement. Look up ring localization.
    - dbranes 8y ago
      (the embedding is full & faithful)
    - NotAnEconomist 8y ago
      > Hence the equivalence I claimed is a precise statement 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.