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Ho, Weng Kin
The notion of formal balls was first introduced by Weihrauch and Schreiber in 1981. It was first used to provide an environment for which metric spaces can be embedded. In 1998, Edalat and Heckmann demonstrated how important this notion is. They found that the complete metric spaces are exactly the metric spaces whose posets of formal balls are domains. This result, which we call the Edalat-Heckmann Theorem, is regarded by several researchers as one of the most important results in domain theory.
In this thesis, we investigate some further applications of formal balls in domain theory. We find that there are various weaker conditions (than the Edalat-Heckmann Theorem) on the formal balls to characterize complete metric spaces. We also strengthen an existing result by Kostanek and Waszkiewicz by showing that continuous Yoneda quasi-metric spaces are exactly the quasi-metric spaces whose posets of formal balls are domains.
We propose a notion of quasi-continuous Yoneda complete quasi-metric space. We then demonstrate the validity of this notion, by showing its confluence with the other well-known results in the literature. In particular, we show that the quasi-continuous Yoneda complete quasi-metric spaces are exactly the Yoneda complete quasi-metric spaces whose posets of formal balls are quasi-continuous dcpos possessing certain properties. We propose a notion of finitely-generated maps of quasi-metric spaces. By passing through the passage of formal balls, we show that a Yoneda complete quasi-metric space is quasi-continuous if and only if the hyperspace of finitely-generated maps is continuous. This extends a result due to Heckmann and Keimel, which states that a dcpo is quasi-continuous if and only if the set of finitely-generated subsets ordered by reverse inclusion is continuous.
Finally, we propose an idempotent and categorical Yoneda completion to any quasi-metric space, which is done using a modified dcpo completion on its poset of formal balls. A pleasant consequence of this is that the category of Yoneda complete quasi-metric spaces with the Y -continuous maps is a reflective subcategory of the category of quasi-metric spaces and the Y -continuous maps.
|Appears in Collections:||Doctor of Philosophy (Ph.D.)|
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