- #1
Parag Kulkarni
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Hi,
How and why set of natural numbers is closed?
How and why set of natural numbers is closed?
Good question. I think it's about metric space.micromass said:Closed in what topological/metric space?
What are your thoughts?
I assume you are replying to my post. There is no assumed embedding of the naturals into the generic metric space.HallsofIvy said: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.
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.WWGD said:I assume you are replying to my post. There is no assumed embedding of the naturals into the generic metric space.
The closed set of natural numbers, also known as the set of whole numbers, is a set of positive integers that starts from 0 and continues infinitely.
The closed set of natural numbers is important because it forms the foundation for basic arithmetic and mathematical operations. It also helps in understanding the concept of infinity and is used in various fields of science and mathematics.
The closed set of natural numbers includes 0, while the open set does not. Additionally, the closed set is a finite set, while the open set is an infinite set.
No, the closed set of natural numbers is considered the highest possible set of positive integers and cannot be extended beyond infinity. It is a concept that goes beyond human comprehension.
The closed set of natural numbers is used in various real-life situations, such as counting objects, measuring quantities, and keeping track of time. It is also essential in fields like economics, physics, and computer science.