We introduce and discuss various properties of sequences of subsets {On} of metric spaces with the property that the limit of delta(On} ) is 0 where delta denotes the diameter of a set, which we call sequentially decreasing subsets. As applications of the theory developed, we give a short proof of a well known necessary condition for a metric space to be connected, give sufficient conditions for subsets of a connected metric space to be totally disconnected, and discuss a specific outer measure on metric spaces.