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In [[mathematics]], a '''totally bounded set''' is any [[set|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 [[bounded set|boundedness]]. | In [[mathematics]], a '''totally bounded set''' is any [[set|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 [[bounded set|boundedness]]. | ||
==Formal definition== | ==Formal definition== | ||
Let ''X'' be a metric space. A set <math>A \subset X</math> is totally bounded if for any | Let ''X'' be a metric space. A set <math>A \subset X</math> is totally bounded if for any [[real number]] ''r''>0 there exists a finite number ''n''(''r'') (that depends on the value of ''r'') of [[metric space#Metric topology|open balls]] of radius ''r'', <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 \subseteq \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 (topology)|closure]] is [[compact space|compact]]. | |||
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Latest revision as of 16:01, 29 October 2024
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 real number r>0 there exists a finite number n(r) (that depends on the value of r) of open balls of radius r, , with , such that .
Properties
- A subset of a complete metric space is totally bounded if and only if its closure is compact.