It could be that the author was thinking about sequential compactness: every sequence of elements of the space has a convergent subsequence (with its limit also

It could be that the author was thinking about sequential compactness: every sequence of elements of the space has a convergent subsequence (with its limit also in the same space). For metric spaces, sequential and usual compactness coincide: https://en.m.wikipedia.org/wiki/Sequentially_compact_space https://en.m.wikipedia.org/wiki/Sequentially_compact_space