0
$\begingroup$

Any Cauchy net in a metric space is convergent? I know that if (X,d) is metric space COMPLETE then cauchy net in X is convergent in X. But, only (X,d) is metric space (not complete) It is true?

pd: As additional doubt. In general, any succession of Cauchy in a set X is always converges in (probability another) some set in somewhere?

$\endgroup$
  • $\begingroup$ The definition of a complete metric space is precisely that every Cauchy net converges in the metric space, so this fails every time a metric space is not complete. user348279's answer gives an example of a non-complete metric space. $\endgroup$ – Reinstate Monica Nov 21 '18 at 1:02
0
$\begingroup$

No: in particular, the statement is false for sequences, and all sequences are nets (they're those nets for which the directed set is the natural numbers with the obvious order relation). For an explicit example, take any Cauchy sequence of rationals that converges in the reals to $\pi$. In the metric space $\mathbb{Q}$, this sequence does not converge.

I'm not at all sure what you mean by your edit, but I think that you're asking if, for every metric space $X$, there is some complete metric space that includes both $X$ and all limits of $X$. If that's correct, then the answer is yes: you simply take all of those things and throw them together, and you're done: this is called the completion of $X$, and is, for example, one of the standard definitions/constructions of $\mathbb{R}$: it is precisely the completion of the rationals.

$\endgroup$
  • $\begingroup$ I do indeed, oops. $\endgroup$ – user3482749 Nov 21 '18 at 0:52

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.