Rocky Mountain Journal of Mathematics

On topological spaces that have a bounded complete DCPO model

Zhao Dongsheng and Xi Xiaoyong

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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.

Article information

Rocky Mountain J. Math., Volume 48, Number 1 (2018), 141-156.

First available in Project Euclid: 28 April 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 06B30: Topological lattices, order topologies [See also 06F30, 22A26, 54F05, 54H12] 06B35: Continuous lattices and posets, applications [See also 06B30, 06D10, 06F30, 18B35, 22A26, 68Q55] 54A05: Topological spaces and generalizations (closure spaces, etc.)

Scott topology maximal point space bounded complete dcpo model CK-open set CK-filter defined space


Dongsheng, Zhao; Xiaoyong, Xi. On topological spaces that have a bounded complete DCPO model. Rocky Mountain J. Math. 48 (2018), no. 1, 141--156. doi:10.1216/RMJ-2018-48-1-141.

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  • M. Ali-Akbaria, B. Honarib and M. Pourmahdiana, Any $T_1$ space has a continuous poset model, Topol. Appl. 156 (2009), 2240–2245.
  • A. Edalat and R. Heckmann, A computational model for metric spaces, Theor. Comp. Sci. 193 (1998), 53–73.
  • R. Engelking, General topology, Heldermann Verlag, Berlin, 1989.
  • M. Erné, Algebraic models for $T_1$-spaces, Topol. Appl. 158 (2011), 945–962.
  • S.P. Franklin, Spaces in which sequences suffice, Fund. Math. 57 (1965), 107–115.
  • ––––, Spaces in which sequences suffice, II, Fund. Math. 61 (1967), 51–56.
  • G. Gierz, K.H. Hofmann, K. Keimel, et al., Continuous lattices and domains, Volume 93, Cambridge University Press, Cambridge, 2003.
  • J. Goubault-Larrecq, Non-Hausdorff topology and domain theory: Selected topics in point-set topology, Volume 22, Cambridge University Press, Cambridge, 2013.
  • 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.
  • R. Kopperman, A. Künzi and P. Waszkiewicz, Bounded complete models of topological spaces, Topol. Appl. 139 (2004), 285–297.
  • J.D. Lawson, Spaces of maximal points, Math. Struct. Comp. Sci. 7 (1997), 543–555.
  • ––––, Computation on metric spaces via domain theory, Topol. Appl. 85 (1998), 247–263.
  • L. Liang and K. Klause, Order environment of topological spaces, Acta Math. Sinica 20 (2004), 943–948.
  • K. Martin, Ideal models of spaces, Theor. Comp. Sci. 305 (2003), 277–297.
  • ––––, Domain theoretic models of topological spaces, Electr. Notes Theor. Comp. Sci. 13 (1998), 173–181.
  • ––––, Nonclassical techniques for models of computation, Topol. Proc. 24 (1999), 375–405.
  • ––––, The regular spaces with countably based models, Theor. Comp. Sci. 305 (2003), 299–310.
  • D.S. Scott, Continuous lattices, in Toposes, algebraic geometry and logic, Lect. Notes Math. 274, Springer-Verlag, New York, 1972.
  • P. Waszkiewicz, How do domains model topologies?, Electr. Notes Theor. Comp. Sci. 83 (2004).
  • K. Weihrauch and U. Schreiber, Embedding metric spaces into cpo's, Theor. Comp. Sci. 16 (1981), 5–24.
  • X. Xi and D. Zhao, Well-filtered spaces and their dcpo models, Math. Struct. Comp. Sci. 27 (2017), 507–515.
  • D. Zhao, Poset models of topological spaces, in Proc. Inter. Conf. Quant. Logic Quantification of Software, Global-Link Publisher, 2009.
  • D. Zhao and X. Xi, Dcpo models of $T_1$ topological spaces, Math. Proc. Cambr. Philos. Soc. 164 (2018), 125–134.