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.
> 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:
> 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.
In the usual mathematical sense of the words you are using, topologies aren’t even the right type of object to admit a notion of continuity. Your statement doesn’t even make sense. It’s maps between them that can be continuous.
In fact, a topological space is sort of the minimal amount of structure a set needs to have to be able to talk about continuity of maps to/from it.
It is not always done, but it is still correct, to replace the objects of a category with the identity morphisms. So in Top, it is totally correct to think of topological space as the identity homeomorphism, which is indeed continuous.
I'm aware; in mathematics it's possible to replace almost anything with some other thing to make the statement you want to be true come true. But it's usually just gonna confuse everyone.
Well the thing I chose to replace the thing with is actually isomorphic (type equivalent) to the thing I replaced. So that's quite a bit more constrained than "replacing anything with anything". Not only are the arrows the only thing that matters, but its cleaner to suppose that they're the only thing there is.
You and OP are using the word continuous in two different contexts. Generally one would not say that the integers with the trivial topology is continuous. It’s a discrete space with a topology. But when someone says a space is continuous generally they mean not discrete.
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.
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.
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.
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.