Date of Final Oral Examination (Defense)
Type of Culminating Activity
Master of Science in Mathematics
Zachariah Teitler, Ph.D.
Jens Harlander, Ph.D.
Samuel Coskey, Ph.D.
A point x is a density point of a set A if all of the points except a measure zero set near to x are contained in A. In the usual topology on ℝ, a set is open if shrinking intervals around each point are eventually contained in the set. The density topology relaxes this requirement. A set is open in the density topology if for each point, the limit of the measure of A contained in shirking intervals to the measure of the shrinking intervals themselves is one. That is, for any point x and a small enough interval Ix, Ix has measure in A arbitrarily close to the measure of Ix. If x has property (1), it is a density point of A.
The density topology is a refinement of the usual topology. As such, it inherits many topological properties from the usual topology. The topology is both Hausdorff and completely regular. This paper will define the density topology starting from Lebesgue measure. After defining the topology, we will demonstrate topological properties including separation and connectedness properties. The density function is related to the topological operations of interior and closure. In addition, the Lebesgue measurable sets are precisely the Borel sets in the density topology.
The density topology can be defined on any space that has a Lebesgue measure and for which the Lebesgue Density Theorem holds. The topology is easily defined on the Cantor space, but is more difficult to define on the space of continuous functions C[0,1]. We explore these results in the final chapters, including a cursory introduction to prevalent and shy sets, an infinite-dimensional analogue of the density topology.
Nygard, Stuart, "The Density Topology on the Reals with Analogues on Other Spaces" (2016). Boise State University Theses and Dissertations. 1143.