-1

After reading the theroem about Sequential characterization of closedness of the set, and the definition of a complete set(a metric space is said to be complete if every Cauchy sequence has its limit in the space X ), I can't undertand what's the difference between both claimings.

Andrés E. Caicedo
  • 79,201
00strich
  • 35
  • 1
    It's hard to guess what statements you are referring to. In order for your question to be answerable, you should include the full statements. – Lee Mosher Feb 14 '18 at 17:52

1 Answers1

0

My guess is that the claims are these:

  • $S$ is closed if whenever a sequence $(x_n)_{n\in\mathbb N}$ of elements of $S$ converges, $\lim_nx_n\in S$;
  • $S$ is complete if every Cauchy sequence of elements of $S$ converges to an element of $S$.

The difference is that in the case of closed sets we are assuming that the sequence is convergent, whereas in the case of complete sets we are assuming that the sequence is a Cauchy sequence. These are distinct assumptions.

José Carlos Santos
  • 427,504
  • Yes, but I think that Cauchy sequences and convergent sequences are the same, aren't they? – 00strich Feb 14 '18 at 18:05
  • 1
    @Leire In a non-complete metric space, they're different. For example, in $(0, 1)$, the sequence $a_n = \frac{1}{n}$ is Cauchy but it has no limit in the metric space $(0, 1)$. – Daniel Schepler Feb 14 '18 at 18:07
  • There is a theorem that gives a relationship, though: A subset of a complete metric space is closed if and only if it is complete. – Daniel Schepler Feb 14 '18 at 18:08
  • 1
    Worth also noting that closeness is a relative property (of $S$ within whatever metric space $X\supset S$ you are looking at) - you cannot decide if $S$ is closed without knowing $X$. (For example, if you take $S$ itself as a metric space, it is always closed "in itself".) On the other hand, completeness is an intrinsic property of $S$ - doesn't matter whether it is taken "in itself" or as a subset of a bigger metric space. –  Feb 14 '18 at 18:29
  • @user8734617 That's a nice remark. That is connected to the fact that, given a sequence, being a Cauchy sequence is an intrinsic property, whereas being convergent is not. – José Carlos Santos Feb 14 '18 at 18:31