From my own definition, I have concluded that a complete metric space is a set and a metric where the set consists of no holes in it. Book definitions describe that "A complete metric space is a metric space in which every Cauchy sequence is convergent." I understand that a metric is a distance measuring device defined on an arbitrary set, and when speaking of a "metric space" they are talking about a set and a metric defined on that set (X,d). However, I have yet to get an understanding of a "Cauchy sequence" when speaking of a Complete Metric Space. I am seeking an example of a complete metric space, relatively one that I can interpret.
[Math] Understanding Complete Metric Spaces and Cauchy Sequences
functional-analysisgeneral-topologymetric-spacessequences-and-series
Best Answer
Example -1: Any set endowed with the discrete metric is complete: every Cauchy sequence is eventually constant, hence convergent.
Example 0: A subset $Y$ of a complete metric space $(X,d)$ is complete with the inherited metric if and only if it is closed.
Example 1: The real numbers $\mathbb{R}$ with $d(x,y) = |x-y|$. (Some people regard using $\mathbb{R}$ as an early example of a metric space to be circular; I am not one of them.)
Example 2: Any compact metric space. (More generally, one has the characterization of compact metric spaces as those which are complete and totally bounded.)
Example 3: a) For any positive integer $n$, if $(X_1,d_1),...,(X_n,d_n)$ are complete metric spaces, and we endow the Cartesian product $X = \prod_{i=1}^n X_i$ with any of several reasonable metrics -- e.g. $d(x,y) = \max_{1 \leq i \leq n} d(x_i,y_i)$ -- then $(X,d)$ is a complete metric space.
b) If $\{(X_n,d_n)\}_{n=1}^{\infty}$ is a sequence of complete metric spaces, and we endow $X = \prod_{i=1}^{\infty} X_i$ with the metric $d(x,y) = \sum_{i=1}^{\infty} \frac{1}{2^i} \frac{ d_i(x_i,y_i)}{1+d_i(x_i,y_i)}$, then $(X,d)$ is a complete metric space.
Example 4: For any metric space $X$, let $C_b(X)$ be the set of bounded, continuous functions $f: X \rightarrow \mathbb{R}$, endowed with the metric $d(f,g) = \sup_{x \in X} |f(x) - g(x)|$. This is a complete metric space and indeed a Banach space.
Example 5: The completion of any metric space. For instance, completing the rational numbers with respect to the $p$-adic metric one gets the field $\mathbb{Q}_p$ of p-adic numbers.
I thought about taking seriously the idea of formalizing "no holes" as a definition of a complete metric space. Here is what I came up with:
Proposition: For a metric space $(X,d)$, the following are equivalent:
(i) For any isometric embedding $\iota: (X,d) \rightarrow (Y,d)$ of $X$ into another metric space $Y$ and any sequence $\{x_n\}$ in $X$, if $\iota(x_n)$ converges in $Y$ then $x_n$ converges in $X$.
(ii) $X$ is complete.
Proof: The basic observations here are that if $\iota: (X,d) \rightarrow (Y,d)$ is an isometric embedding and $\{x_n\}$ is a sequence in $X$, then:
$\bullet$ $\{x_n\}$ is Cauchy iff $\{\iota(x_n)\}$ is Cauchy, hence also
$\bullet$ if $\{ \iota(x_n)\}$ is convergent, then $\{x_n\}$ is Cauchy.
Then (ii) $\implies$ (i) is immediate; to show (i) $\implies$ (ii) look at the completion $\iota: X \rightarrow \tilde{X}$ of $X$.
Thus the "holes" in $X$ are detected by embeddings into larger spaces. I am skeptical though that this definition would be helpful for beginning students: aside from relying on the existence of the completion of a metric space, the idea of considering all possible embeddings of one metric space into another seems relatively abstract and sophisticated.