Totally bounded set

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 $A\subset X$ 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, $B_{r}(x_{1}),\ldots ,B_{r}(x_{n(r)})\,$, with $x_{1},\ldots ,x_{n(r)}\in X$, such that $A\subseteq \cup _{k=1}^{n(r)}B_{r}(x_{k})$.

Properties

• A subset of a complete metric space is totally bounded if and only if its closure is compact.