Let $(M, d)$ be a metric space. I have a conjecture that is something along the lines of: If $M$ is closed, then it is complete, and was wondering how true this statement is.
I know that it might not be sensible to say that a metric space $M$ is closed; the property of being closed is usually relative to a surrounding space, but I think I can reformulate this statement in two ways that might make sense.
1) If $M$ is closed in any metric space $M'$ that contains $M$, then $M$ is complete
2) If $M$ contains all of its limit points, then $M$ is complete.
The idea is that a Cauchy sequence should be convergent and the only situations where it is not are when the limit is outside of the metric space. Is this statement true? If not, I would really appreciate some counterexamples.
If it is true, why is the completion defined with equivalence classes when instead we can say the completion of a metric space $M$ is just it's closure $\overline{M}$? I am definitely missing something, so would love some clarification!
Note: I have not done too much Topology so would appreciate if we could keep the answer to metric spaces.
Statement 1 is correct. Completeness is equivalent to "closed in any larger metric space $M'$" (with the understanding that the two metrics agree on $M$).
Indeed, every metric space has a completion $\widehat{M}$, which is a complete metric space containing $M$ as a dense subset. By assumption, $M$ is closed as a subset of $\widehat{M}$. Being both closed and dense implies that $M=\widehat{M}$. Thus $M$ is complete.
Statement (2) does not really make sense. There is no concept of a limit point being just "outside", not contained in anything. In order to talk about limit points that are not in $M$, we need a larger space $M'$ in which those points are contained. And then we arrive back at statement (1).
For the same reason, we don't get anything new from "the closure of $M$" without specifying a larger space in which the closure is taken.