## On topological spaces that have a bounded complete dcpo model.(English)Zbl 1427.06001

Summary: A dcpo model of a topological space $$X$$ is a dcpo (directed complete poset) $$P$$ such that $$X$$ is homeomorphic to the maximal point space of $$P$$ with the subspace topology of the Scott space of $$P$$. It has been previously proved by Xi and Zhao that every $$T_1$$ space has a dcpo model. It is, however, still unknown whether every $$T_1$$ space has a bounded complete dcpo model (a poset is bounded complete if each of its upper bounded subsets has a supremum). In this paper, we first show that the set of natural numbers equipped with the co-finite topology does not have a bounded complete dcpo model and then prove that a large class of topological spaces (including all Hausdorff $$k$$-spaces) have a bounded complete dcpo model. We shall mainly focus on the model formed by all of the nonempty closed compact subsets of the given space.

### MSC:

 06B35 Continuous lattices and posets, applications 06B30 Topological lattices 54A05 Topological spaces and generalizations (closure spaces, etc.) 54D10 Lower separation axioms ($$T_0$$–$$T_3$$, etc.)
Full Text:

### References:

 [1] M. Ali-Akbaria, B. Honarib and M. Pourmahdiana, Any $$T_1$$ space has a continuous poset model, Topol. Appl. 156 (2009), 2240-2245. · Zbl 1173.54010 [2] A. Edalat and R. Heckmann, A computational model for metric spaces, Theor. Comp. Sci. 193 (1998), 53-73. · Zbl 1011.54026 [3] R. Engelking, General topology, Heldermann Verlag, Berlin, 1989. [4] M. Erné, Algebraic models for $$T_1$$-spaces, Topol. Appl. 158 (2011), 945-962. · Zbl 1216.54002 [5] S.P. Franklin, Spaces in which sequences suffice, Fund. Math. 57 (1965), 107-115. · Zbl 0132.17802 [6] —-, Spaces in which sequences suffice, II, Fund. Math. 61 (1967), 51-56. · Zbl 0168.43502 [7] G. Gierz, K.H. Hofmann, K. Keimel, et al., Continuous lattices and domains, Volume 93, Cambridge University Press, Cambridge, 2003. · Zbl 1088.06001 [8] J. Goubault-Larrecq, Non-Hausdorff topology and domain theory: Selected topics in point-set topology, Volume 22, Cambridge University Press, Cambridge, 2013. · Zbl 1280.54002 [9] K.H. Hofmann and J.D. Lawson, On the order-theoretical foundation of a theory of quasicompactly generated spaces without separation axiom, J. Australian Math. Soc. 36 (1984), 194-212. · Zbl 0536.06005 [10] R. Kopperman, A. Künzi and P. Waszkiewicz, Bounded complete models of topological spaces, Topol. Appl. 139 (2004), 285-297. · Zbl 1054.06004 [11] J.D. Lawson, Spaces of maximal points, Math. Struct. Comp. Sci. 7 (1997), 543-555. · Zbl 0985.54025 [12] —-, Computation on metric spaces via domain theory, Topol. Appl. 85 (1998), 247-263. · Zbl 0922.54025 [13] L. Liang and K. Klause, Order environment of topological spaces, Acta Math. Sinica 20 (2004), 943-948. · Zbl 1072.54021 [14] K. Martin, Ideal models of spaces, Theor. Comp. Sci. 305 (2003), 277-297. · Zbl 1044.54005 [15] —-, Domain theoretic models of topological spaces, Electr. Notes Theor. Comp. Sci. 13 (1998), 173-181. [16] —-, Nonclassical techniques for models of computation, Topol. Proc. 24 (1999), 375-405. · Zbl 1029.06501 [17] —-, The regular spaces with countably based models, Theor. Comp. Sci. 305 (2003), 299-310. · Zbl 1053.54037 [18] D.S. Scott, Continuous lattices, in Toposes, algebraic geometry and logic, Lect. Notes Math. 274, Springer-Verlag, New York, 1972. [19] P. Waszkiewicz, How do domains model topologies?, Electr. Notes Theor. Comp. Sci. 83 (2004). [20] K. Weihrauch and U. Schreiber, Embedding metric spaces into cpo’s, Theor. Comp. Sci. 16 (1981), 5-24. · Zbl 0485.68040 [21] X. Xi and D. Zhao, Well-filtered spaces and their dcpo models, Math. Struct. Comp. Sci. 27 (2017), 507-515. · Zbl 1378.06005 [22] D. Zhao, Poset models of topological spaces, in Proc. Inter. Conf. Quant. Logic Quantification of Software, Global-Link Publisher, 2009. [23] D. Zhao and X. Xi, Dcpo models of $$T_1$$ topological spaces, Math. Proc. Cambr. Philos. Soc. 164 (2018), 125-134. · Zbl 1469.06012
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.