proof that a metric space is compact if and only if it is complete and totally bounded
Theorem^{}: A metric space is compact^{} if and only if it is complete^{} and totally bounded^{}.
Proof. Let $X$ be a metric space with metric $d$. If $X$ is compact, then it is sequentially compact and thus complete. Since $X$ is compact, the covering of $X$ by all $\u03f5$-balls must have a finite subcover, so that $X$ is totally bounded.
Now assume that $X$ is complete and totally bounded. For metric spaces, compact and sequentially compact are equivalent^{}; we prove that $X$ is sequentially compact. Choose a sequence^{} ${p}_{n}\in X$; we will find a Cauchy subsequence^{} (and hence a convergent^{} subsequence, since $X$ is complete).
Cover $X$ by finitely many balls of radius $1$ (since $X$ is totally bounded). At least one of those balls must contain an infinite^{} number of the ${p}_{i}$. Call that ball ${B}_{1}$, and let ${S}_{1}$ be the set of integers $i$ for which ${p}_{i}\in {B}_{1}$.
Proceeding inductively, it is clear that we can define, for each positive integer $k>1$, a ball ${B}_{k}$ of radius $1/k$ containing an infinite number of the ${p}_{i}$ for which $i\in {S}_{k-1}$; define ${S}_{k}$ to be the set of such $i$.
Each of the ${S}_{k}$ is infinite, so we can choose a sequence ${n}_{k}\in {S}_{k}$ with $$ for all $k$. Since the ${S}_{k}$ are nested, we have that whenever $i,j\ge k$, then ${n}_{i},{n}_{j}\in {S}_{k}$. Thus for all $i,j\ge k$, ${p}_{{n}_{i}}$ and ${p}_{{n}_{j}}$ are both contained in a ball of radius $1/k$. Hence the sequence ${p}_{{n}_{k}}$ is Cauchy.
References
- 1 J. Munkres, Topology^{} , Prentice Hall, 1975.
Title | proof that a metric space is compact if and only if it is complete and totally bounded |
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Canonical name | ProofThatAMetricSpaceIsCompactIfAndOnlyIfItIsCompleteAndTotallyBounded |
Date of creation | 2013-03-22 18:01:06 |
Last modified on | 2013-03-22 18:01:06 |
Owner | rm50 (10146) |
Last modified by | rm50 (10146) |
Numerical id | 5 |
Author | rm50 (10146) |
Entry type | Theorem |
Classification | msc 40A05 |
Classification | msc 54D30 |