5 ms·

How is this fundamentally different than considering data structures as graphs?

anon291 3y ago
Graphs are discrete, topologies are potentially continuous. Moreover, you can do different things with them such as create homeomorphisms to another topology much more easily than you can create bijections between graphs. In general, continuity lets you assume things that are impossible in discrete spaces. For example, many optimization problems are really easy in continuous spaces but really hard in discrete ones (linear programming for example) Given the recent success with vectors as a general model for data (as witnessed by the continued success with deep neural networks), it's an interesting discussion to have.
- dlahoda 3y ago
  feels so true.
- StevenXC 3y ago
  Also relevant: https://en.m.wikipedia.org/wiki/Topological_data_analysis https://en.m.wikipedia.org/wiki/Topological_data_analysis
- JensW 3y ago
  Discrete spaces can also be topological spaces, see discrete topology
  - HighFreqAsuka 3y ago
    Did you not read the word “potentially”? Topological spaces are a more general case of spaces that contain the discrete case as a subset.
- aleph_minus_one 3y ago
  > Moreover, you can do different things with them such as create homeomorphisms to another topology much more easily than you can create bijections between graphs. In general, continuity lets you assume things that are impossible in discrete spaces. If you argue with more generality: why not consider sites (and, relatedly, topoi) instead of topological spaces then: > https://en.wikipedia.org/wiki/Grothendieck_topology https://en.wikipedia.org/wiki/Grothendieck_topology > https://en.wikipedia.org/wiki/Topos https://en.wikipedia.org/wiki/Topos
  - anon291 3y ago
    > If you argue with more generality: why not consider sites (and, relatedly, topoi) instead of topological spaces then: I thought linear programming would be something everyone knows, and I am not the original author so I can't speak for why they chose topological spaces instead of anything listed here. I think their e-mails are on the paper. Perhaps e-mailing them will help elucidate their choice.
- mathgradthrow 3y ago
  Topologies are necessarily continuous
  - anon291 3y ago
    Perhaps the confusion is that I should have said topological spaces can be continuous. There are discrete topological spaces. Topologies (which I believe is typically used to refer to the collection of open sets in a topological space) are not functions or relations themselves, so I'm not sure a useful notion of continuity applies there, but if I'm wrong, please inform.
    - m3ndax 3y ago
      There isn't really such a thing as a 'continuous topological space'. Technically speaking, continuity is a property of functions between topological spaces. I think you're being tempted to use the terms continuous and discrete in a more colloquial sense mapping more to uncountable vs countable/countable and finite perhaps. But yeah, you really wouldn't use the term continuous to describe a topological space or a topology.
      - nyrikki 3y ago
        The classic middle-thirds Cantor Set being a topologically set is one of the easiest counter examples to the above misconception that the sets need to be continuous themselves. Being able to define a neighborhood or a concept of closeness is required, but the concept of distance is not required. If you can define a distance a topological space is a metric space If it is locally euclidean it may be a manifold. Really the union and finite intersection of subsets is the formal way of showing something is a topological space. Too har do describe here but that is where the concept of continuity arises.
      - anon291 3y ago
        Connected or complete then.
  - bmacho 3y ago
    This is not even false.
- Kalanos 3y ago
  so topologies are grids (e.g. coordinates) in this case? it's not a great naming choice as "topology" is frequently used to describe network/graph

Graphs are discrete, topologies are potentially continuous. Moreover, you can do different things with them such as create homeomorphisms to another topology much more easily than you can create bijections between graphs. In general, continuity lets you assume things that are impossible in discrete spaces. For example, many optimization problems are really easy in continuous spaces but really hard in discrete ones (linear programming for example) Given the recent success with vectors as a general model for data (as witnessed by the continued success with deep neural networks), it's an interesting discussion to have.

dlahoda 3y ago
feels so true.
StevenXC 3y ago
Also relevant: https://en.m.wikipedia.org/wiki/Topological_data_analysis https://en.m.wikipedia.org/wiki/Topological_data_analysis
JensW 3y ago
Discrete spaces can also be topological spaces, see discrete topology
- HighFreqAsuka 3y ago
  Did you not read the word “potentially”? Topological spaces are a more general case of spaces that contain the discrete case as a subset.
aleph_minus_one 3y ago
> Moreover, you can do different things with them such as create homeomorphisms to another topology much more easily than you can create bijections between graphs. In general, continuity lets you assume things that are impossible in discrete spaces. If you argue with more generality: why not consider sites (and, relatedly, topoi) instead of topological spaces then: > https://en.wikipedia.org/wiki/Grothendieck_topology https://en.wikipedia.org/wiki/Grothendieck_topology > https://en.wikipedia.org/wiki/Topos https://en.wikipedia.org/wiki/Topos
- anon291 3y ago
  > If you argue with more generality: why not consider sites (and, relatedly, topoi) instead of topological spaces then: I thought linear programming would be something everyone knows, and I am not the original author so I can't speak for why they chose topological spaces instead of anything listed here. I think their e-mails are on the paper. Perhaps e-mailing them will help elucidate their choice.
mathgradthrow 3y ago
Topologies are necessarily continuous
- anon291 3y ago
  Perhaps the confusion is that I should have said topological spaces can be continuous. There are discrete topological spaces. Topologies (which I believe is typically used to refer to the collection of open sets in a topological space) are not functions or relations themselves, so I'm not sure a useful notion of continuity applies there, but if I'm wrong, please inform.
  - m3ndax 3y ago
    There isn't really such a thing as a 'continuous topological space'. Technically speaking, continuity is a property of functions between topological spaces. I think you're being tempted to use the terms continuous and discrete in a more colloquial sense mapping more to uncountable vs countable/countable and finite perhaps. But yeah, you really wouldn't use the term continuous to describe a topological space or a topology.
    - nyrikki 3y ago
      The classic middle-thirds Cantor Set being a topologically set is one of the easiest counter examples to the above misconception that the sets need to be continuous themselves. Being able to define a neighborhood or a concept of closeness is required, but the concept of distance is not required. If you can define a distance a topological space is a metric space If it is locally euclidean it may be a manifold. Really the union and finite intersection of subsets is the formal way of showing something is a topological space. Too har do describe here but that is where the concept of continuity arises.
    - anon291 3y ago
      Connected or complete then.
- bmacho 3y ago
  This is not even false.
Kalanos 3y ago
so topologies are grids (e.g. coordinates) in this case? it's not a great naming choice as "topology" is frequently used to describe network/graph