Closed Set

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1. Nov 4, 2015

Parag Kulkarni

Hi,
How and why set of natural numbers is closed?

2. Nov 4, 2015

micromass

Staff Emeritus
Closed in what topological/metric space?

What are your thoughts?

3. Nov 4, 2015

Parag Kulkarni

Good question. I think it's about metric space.

4. Nov 4, 2015

micromass

Staff Emeritus
What metric space?

5. Nov 4, 2015

WWGD

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?

6. Nov 4, 2015

Staff: Mentor

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.

7. Nov 4, 2015

HallsofIvy

Staff Emeritus
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.

8. Nov 4, 2015

WWGD

I assume you are replying to my post. There is no assumed embedding of the naturals into the generic metric space.

9. Nov 4, 2015

Staff: Mentor

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.

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