Domain representations of spaces of compact subsets. (English) Zbl 1197.06003
Summary: We present a method for constructing from a given domain representation of a topological space \(X\) with underlying domain \(D\), a domain representation of a subspace of compact subsets of \(X\) where the underlying domain is the Plotkin powerdomain of \(D\). We show that this operation is functorial over a category of domain representations with a natural choice of morphisms. We study the topological properties of the space of representable compact sets and isolate conditions under which all compact subsets of \(X\) are representable. Special attention is paid to admissible representations and representations of metric spaces.
MSC:
| 06B35 | Continuous lattices and posets, applications |
| 54B30 | Categorical methods in general topology |
| 54D30 | Compactness |
| 54E35 | Metric spaces, metrizability |
Keywords:
domain representation of a topological space; subspace of compact subsets; Plotkin powerdomain; representable compact sets; metric spacesReferences:
| [1] | Smyth, Springer-Verlag Lecture Notes in Computer Science 154 pp 662– (1983) · doi:10.1007/BFb0036946 |
| [2] | Stoltenberg-Hansen, Handbook of Logic in Computer Science 4 pp 357– (1995) |
| [3] | Stoltenberg-Hansen, Mathematical Theory of Domains (1994) · Zbl 0828.06001 · doi:10.1017/CBO9781139166386 |
| [4] | DOI: 10.1016/S0304-3975(99)00045-6 · Zbl 0949.68096 · doi:10.1016/S0304-3975(99)00045-6 |
| [5] | DOI: 10.1016/S0304-3975(98)00282-5 · Zbl 0916.68092 · doi:10.1016/S0304-3975(98)00282-5 |
| [6] | DOI: 10.1016/S0168-0072(96)00017-6 · Zbl 0867.03014 · doi:10.1016/S0168-0072(96)00017-6 |
| [7] | DOI: 10.1016/j.entcs.2006.04.005 · doi:10.1016/j.entcs.2006.04.005 |
| [8] | Normann, Springer-Verlag Lecture Notes in Computer Science 811 (1980) |
| [9] | Abramsky, Handbook of Logic in Computer Science pp 1– (1994) |
| [10] | Kreisel, Constructivity in Mathematics pp 101– (1959) |
| [11] | Kleene, Constructivity in Mathematics pp 81– (1959) |
| [12] | DOI: 10.1017/CBO9780511542725 · Zbl 1088.06001 · doi:10.1017/CBO9780511542725 |
| [13] | Escardó, Logical Methods in Computer Science 4 pp 1– (2008) |
| [14] | Weihrauch, An Introduction to Computable Analysis (2000) |
| [15] | DOI: 10.1016/j.topol.2004.02.011 · Zbl 1066.54028 · doi:10.1016/j.topol.2004.02.011 |
| [16] | DOI: 10.1016/S0049-237X(09)70439-2 · doi:10.1016/S0049-237X(09)70439-2 |
| [17] | DOI: 10.1017/S0960129508007093 · Zbl 1166.03020 · doi:10.1017/S0960129508007093 |
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