Master of Science

In this thesis, we perform an in-depth study of the categorical dualities and equivalences that describe crucial connections between order and topology. The goal of this thesis, as the title indicates, is to study the topological aspects of partial orders. Explicitly, we study how a partially ordered set can be recovered by defining suitable topologies on it. We focus on those recovery processes that can be organized using a categorical framework, which in turn set the stage for us to deal with two main kinds of results. The first kind is the Stone Duality type of results which illustrates the relationship between certain subcategories of distributive lattices and that of topological spaces. The second kind is the Hofmann Duality type of results that connects between the categories of algebraic domains and that of sup- semilattices.

With regards to the organization of content, Part I of this thesis is where we de ne all required foundational tools. Part II is devoted to organizing known results of the first kind in a categorical framework, while Part III presents some original results that can be classified as dualities (or equivalences) of the second kind.