Complex sentence: This definition is well-suited in the case where X is not Hausdorff, and coincides with the usual definition if X is second countable or if every compact saturated subset is closed (which is the case in particular if X is Hausdorff).
Split into simpler sentences: This definition is well-suited for applications in the case where X is not Hausdorff. It coincides with the usual definition if X is second countable or if every compact saturated subset is closed (which is the case in particular if X is Hausdorff).