We introduce and discuss various properties of sequences of
subsets {O_{n}} of metric spaces with
the property that the limit of delta(O_{n}} ) 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.