Pre-Hausdorff objects in topological categories

Date of Award




Degree Name

Doctor of Philosophy (Ph.D.)



First Committee Member

Marvin V. Mielke, Committee Chair


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.



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