is a type of mathematical space
is a type of metric space that is totally bounded
a metric space (𝑋,𝑑) is totally bounded if and only if for every real number 𝜀>0, there exists a FINITE collection of open balls of radius 𝜀 whose centers lie in 𝑋 and whose union contains 𝑋
equivalently, a metric space (𝑋,𝑑) is totally bounded if and only if for every real number 𝜀>0, there exists a FINITE cover such that the radius of each element of the cover is at most 𝜀
any subset of a totally bounded metric space is a bounded set
every totally bounded metric space is also a bounded metric space

Totally Bounded Metric Spaces - Examples

[0, 1]² is a totally bounded space because, for every 𝜀>0, the unit square can be covered by finitely many open discs of radius 𝜀.
[0,1]³ is a totally bounded metric space
[0,1]^𝑛 of any finite dimension is a totally bounded metric space

Totally Bounded Metric Spaces - Non-Examples