#### Presentation Title

Sub-Hausdorff Spaces, The Dual Closure, and Incidence

#### Faculty Mentor

Dr. Alfonso Agnew

#### Start Date

18-11-2017 1:30 PM

#### End Date

18-11-2017 1:45 PM

#### Location

9-285

#### Session

Physical Sciences 2

#### Type of Presentation

Oral Talk

#### Subject Area

physical_mathematical_sciences

#### Abstract

Point-set topology underlies much of modern pure and applied mathematics today. The majority of topological spaces that are studied are Hausdorff, meaning that every two points in the space can be separated by open sets. In this article we study spaces which lack this property, which we call sub-Hausdorff spaces. We develop some tools for studying sub-Hausdorff spaces. In particular, we define the dual closure operator, which in some sense is dual to the well known closure operator in point-set topology. We characterize the properties of the dual closure and present a dual notion of a limit point. We then discuss how this operator characterizes the lack of separability in sub-Hausdorff spaces. Next, we demonstrate how the lack of separability in sub-Hausdorff spaces is connected to the concept of incidence. Finally, we apply these ideas to the Zariski topology of algebraic geometry and show how the dual closure of a point yields the intersection of all closed algebraic sets at that point.

#### Summary of research results to be presented

The discovery of the properties of the dual closure operator are new results. We have discovered deep connections between this operator and sub-Hausdorff spaces, in particular spaces which exhibit T_{0 }separability. This operator allows us to elucidate the properties of points within a space which cannot be separated. In particular, we show that this operator has deep connections to the concept of incidence structure in geometry. We demonstrate that the dual closure operator is able to determine the collection of all closed algebraic sets which intersect at a point in the Zariski topology. This is done by showing that generic points in the Zariski topology correspond to dual limit points contained by the dual closure of a set.

Sub-Hausdorff Spaces, The Dual Closure, and Incidence

9-285

Point-set topology underlies much of modern pure and applied mathematics today. The majority of topological spaces that are studied are Hausdorff, meaning that every two points in the space can be separated by open sets. In this article we study spaces which lack this property, which we call sub-Hausdorff spaces. We develop some tools for studying sub-Hausdorff spaces. In particular, we define the dual closure operator, which in some sense is dual to the well known closure operator in point-set topology. We characterize the properties of the dual closure and present a dual notion of a limit point. We then discuss how this operator characterizes the lack of separability in sub-Hausdorff spaces. Next, we demonstrate how the lack of separability in sub-Hausdorff spaces is connected to the concept of incidence. Finally, we apply these ideas to the Zariski topology of algebraic geometry and show how the dual closure of a point yields the intersection of all closed algebraic sets at that point.