## Dissertations from ProQuest

#### Title

Pre-Hausdorff objects in topological categories

1997

Article

#### Degree Name

Doctor of Philosophy (Ph.D.)

Mathematics

#### First Committee Member

Marvin V. Mielke - Committee Chair

#### Abstract

There are two notions in topological category theory, called preT$\sb2$ and preT$\sp\prime\sb2$, which both reduce to the classical pre-Hausdorff separation axiom in the case of topological spaces. These notions arise in studying the image of a topos in a topological category by the left adjoint of a geometric morphism, which includes as a special case the image of simplicial sets by geometric realization functors. This image is always preT$\sp\prime\sb2$ and, consequently, always preT$\sb2$ as well. This paper deals with the similarities and differences between preT$\sb2$ and preT$\sp\prime\sb2$ objects, as well as the relationship between these and other stuctured objects (such as discrete and indiscrete objects) in a topological category. Typical results: the preT$\sb2$ objects form a topological category, the preT$\sp\prime\sb2$ objects may not; indiscrete objects are preT$\sb2$ but may not be preT$\sp\prime\sb2$; discrete objects and 0-dimensional objects (also defined in this paper) are preT$\sb2$ in a geometric topological category, but may not be preT$\sp\prime\sb2$. New separation axioms for topological spaces are defined and then employed to show why preT$\sp\prime\sb2$ is topological in that case.It is also shown that many familiar full subcategories of topological spaces, including T$\sb0$, T$\sb1$, T$\sb2$, and pre-Hausdorff spaces, are reflective; i.e., their inclusion functors all have a left adjoint. These left adjoints are explicitly constructed, and are then shown to be special cases of general left adjoint constructions in a large class of topological categories, including those over a Grothendieck topos. Throughout the paper, several different characterizations of pre-Hausdorff spaces are established. These include characterizations in terms of the diagonal in a product space (as well as other equivalence relations), Hausdorff separation in certain quotient spaces, and the logic in the topos of sheaves on a topological space. In particular it is shown that if X is a finite space, then the topos of sheaves on X is a Boolean topos if and only if X is pre-Hausdorff.

Mathematics