4 ms·

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.
skhunted 3y ago
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.
- mathgradthrow 3y ago
  If you use the word topology as if you know what you are talking about, you sacrifice this benefit of the doubt.
ducttapecrown 3y ago
You're going to tell me the discrete topology is continuous!?
xanderlewis 3y ago
No, they are not. 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.
- mathgradthrow 3y ago
  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.
  - xanderlewis 3y ago
    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.
    - mathgradthrow 3y ago
      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.

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.

skhunted 3y ago

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.

mathgradthrow 3y ago
If you use the word topology as if you know what you are talking about, you sacrifice this benefit of the doubt.

ducttapecrown 3y ago

You're going to tell me the discrete topology is continuous!?

xanderlewis 3y ago

No, they are not. 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.

mathgradthrow 3y ago
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.
- xanderlewis 3y ago
  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.
  - mathgradthrow 3y ago
    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.