On computational environments of topological spaces

Authors:
Fangping Huang;Jihua Liang
Affiliations:
School of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China;School of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China
Venue:
Theoretical Computer Science
Year:
2008

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Abstract

As has been disclosed by K. Martin, a large number of important topological spaces do not have any continuous domain as their computational model. So it is of interest to study new kinds of pragmatic computational environments so as to model more topological spaces. In this paper we focus on bounded complete continuous posets with enough maximal points, which are shown to be a good choice for computational environments of Tychonoff spaces with no directed complete model. It is proved that the maximal point space of a Choquet complete weak domain is also Choquet complete. Furthermore, it is proved that X is a Tychonoff space iff X has a bounded complete weak domain environment. And it is also shown that Hausdorff compactifications of Tychonoff spaces can be realized via some of their computational environments.