How and why set of natural numbers is closed?
Closed in what topological/metric space?
What are your thoughts?
Good question. I think it's about metric space.
What metric space?
OP:There is no natural definition of natural numbers in a generic metric space. Do you have any particular embedding of the natural numbers in mind?
ℕ as a half-group is closed under addition.
ℕ as a discreet topological space is closed by itself. A discreet metric won't change anything. But both is more or less trivial.
ℕ⊆ℝ is closed since its complement is open, i.e. you can find to each real number r, that is not natural, an open intervall that contains r but still no natural number.
Every topological space is "closed' as a subset of itself. If you have it embedded in the real numbers with the "usual metric", d(x, y)= |x- y|, then it is closed as fresh_42 says.
I assume you are replying to my post. There is no assumed embedding of the naturals into the generic metric space.
Yes, you are right. And I can imagine a couple of very funny embeddings, metric or not. But considering the simplicity of the question it's not very unlikely that ℕ⊂ℝ with it's euclidean metric is meant. And yes, it hasn't been mentioned. Reading the questions here I found that most of them are far from being precise or even clear.
Separate names with a comma.