Totally bounded set: Difference between revisions
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Let ''X'' be a metric space. A set <math>A \subset X</math> is totally bounded if for any radius ''r>0'' the exist a finite number ''n(r)'' (that depends on the value of ''r'') of [[metric space#Metric topology|open balls]] <math>B_r(x_1),\ldots,B_r(x_{n(r)})</math>, with <math>x_1,\ldots,x_{n(r)} \in X</math>, such that <math>A \subset \cup_{k=1}^{n(r)}B_r(x_{k})</math>. | Let ''X'' be a metric space. A set <math>A \subset X</math> is totally bounded if for any radius ''r>0'' the exist a finite number ''n(r)'' (that depends on the value of ''r'') of [[metric space#Metric topology|open balls]] <math>B_r(x_1),\ldots,B_r(x_{n(r)})</math>, with <math>x_1,\ldots,x_{n(r)} \in X</math>, such that <math>A \subset \cup_{k=1}^{n(r)}B_r(x_{k})</math>. | ||
== | ==Properties== | ||
* [[ | * A subset of a [[complete metric space]] is totally bounded if and only if its [[closure (mathematics)|closure]] is [[compact space|compact]]. | ||
Revision as of 12:07, 28 December 2008
In mathematics, a totally bounded set is any subset of a metric space with the property that for any positive radius r>0 it is contained in some union of a finite number of "open balls" of radius r. In a finite dimensional normed space, such as the Euclidean spaces, total boundedness is equivalent to boundedness.
Formal definition
Let X be a metric space. A set is totally bounded if for any radius r>0 the exist a finite number n(r) (that depends on the value of r) of open balls , with , such that .
Properties
- A subset of a complete metric space is totally bounded if and only if its closure is compact.