# Generalized Ultrametric Spaces in Quantitative Domain Theory

Generalized Ultrametric Spaces in Quantitative Domain Theory

**Abstract.** Domains and metric spaces are two central tools for the study of denotational semantics in computer science, but are otherwise very different in many fundamental aspects. A construction that tries to establish links between both paradigms is the space of formal balls, a continuous poset which can be defined for every metric space and that reflects many of its properties. On the other hand, in order to obtain a broader framework for applications and possible connections to domain theory, generalized ultrametric spaces (gums) have been introduced. In this paper, we employ the space of formal balls as a tool for studying these more general metrics by using concepts and results from domain theory. It turns out that many properties of the metric can be characterized via its formal-ball space. Furthermore, we can state new results on the topology of gums as well as two new fixed point theorems, which may be compared to the Prieß-Crampe and Ribenboim theorem, and the Banach fixed point theorem, respectively. Deeper insights into the nature of formal-ball spaces are gained by applying methods from category theory. Our results suggest that, while being a useful tool for the study of gums, the space of formal balls does not provide the hoped-for general connection to domain theory.

*Published at Theoretical Computer Science (Journal paper) *

*Download PDF* *(last update: December 01 2006)*

## Citation details

- Markus Krötzsch. Generalized Ultrametric Spaces in Quantitative Domain Theory. In Theoretical Computer Science 368 (1–2), pp. 30–49. ElsevierProperty "Publisher" has a restricted application area and cannot be used as annotation property by a user. 2006.

author = {Markus Kr{\"o}tzsch},

title = {Generalized Ultrametric Spaces in Quantitative

Domain Theory},

journal = {Theoretical Computer Science},

volume = {368},

number = {1--2},

pages = {30--49},

year = {2006}

}

## Remarks

The above link points to the Technical Report WV-04-02 of the Center for Computational Logic at TU Dresden. The original article is available for subscribers of TCS at [1].