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.

Author Bio

This paper was based on research completed in the summer of 2003 for an REU at New Jersey Institute of Technology (NJIT), under the advisement of Prof. Denis Blackmore. As of fall 2004, I am a senior at NJIT pursuing a B.S. in applied mathematics with a minor in economics. After graduation I plan to pursue a Ph.D. in mathematics with a possible graduate minor in mathematical economics. My non-mathematical interests include politics, art, and music.