Confusion on definition of complete metric spaces

Question

So I am a complete beginner to functional analysis and topology and I recently learned what a complete metric space is I understand that every cauchy sequence must converge to a limit in the set. My confusion is why is the set of all natural numbers $\mathbb{N}$ not a complete metric space. I also don't grasp why a complete metric space is automatically not a closed set because a closed set contains all of its limit points?. This is probably very trivial to you all and I admit my inquiries on the matter may not seem clear but it has really confused me. Thanks in advance

Answer 1

I'm going to expand on Brian Scott's comment. First of all, the natural numbers $\mathbb{N}$ are in fact complete. You can see this because all Cauchy sequences of natural numbers must eventually be constant and therefore converge.

Secondly, you should be aware that while completeness is an internal/absolute property of a metric space, being closed is defined relative to some ambient space. For example, while $(0,1)$ is not closed when considered as a subset of $\mathbb{R}$, it is closed when considered as a subset of itself. That's because $0$ and $1$ are no longer limit points of $(0,1)$ as a subset of $(0,1).$ They can't be because they are not even points of $(0,1).$ Actually, all metric spaces are closed as subsets of themselves.

There is a sense in which your intuition holds. If $X\subseteq Y$ are metric spaces and $X$ is complete then $X$ must be closed. If $Y$ is assumed to be complete, then $X$ is complete if and only if it is closed.