Article
| Title: | Quasicontinuous spaces (English) |
| Author: | Lu, Jing |
| Author: | Zhao, Bin |
| Author: | Wang, Kaiyun |
| Author: | Zhao, Dongsheng |
| Language: | English |
| Journal: | Commentationes Mathematicae Universitatis Carolinae |
| ISSN: | 0010-2628 (print) |
| ISSN: | 1213-7243 (online) |
| Volume: | 63 |
| Issue: | 4 |
| Year: | 2022 |
| Pages: | 513-526 |
| Summary lang: | English |
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| Category: | math |
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| Summary: | We lift the notion of quasicontinuous posets to the topology context, called quasicontinuous spaces, and further study such spaces. The main results are: (1) A $T_{0}$ space $(X,\tau)$ is a quasicontinuous space if and only if $SI(X)$ is locally hypercompact if and only if $(\tau_{SI},\subseteq)$ is a hypercontinuous lattice; (2) a $T_{0}$ space $X$ is an $SI$-continuous space if and only if $X$ is a meet continuous and quasicontinuous space; (3) if a $C$-space $X$ is a well-filtered poset under its specialization order, then $X$ is a quasicontinuous space if and only if it is a quasicontinuous domain under the specialization order; (4) there exists an adjunction between the category of quasicontinuous domains and the category of quasicontinuous spaces which are well-filtered posets under their specialization orders. (English) |
| Keyword: | quasicontinuous space |
| Keyword: | hypercontinuous lattice |
| Keyword: | $SI$-continuous space |
| Keyword: | locally hypercompact space |
| Keyword: | meet continuous space |
| MSC: | 06B30 |
| MSC: | 06B35 |
| MSC: | 54D10 |
| idZBL: | Zbl 07723832 |
| idMR: | MR4577045 |
| DOI: | 10.14712/1213-7243.2023.005 |
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| Date available: | 2023-04-20T13:58:16Z |
| Last updated: | 2023-10-27 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/151650 |
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| Reference: | [1] Adámek J., Herrlich H., Strecker G. E.: Abstract and Concrete Categories.The Joy of Cats, Pure and Applied Mathematics, A Wiley-Interscience Publication, John Wiley & Sons, New York, 1990. MR 1051419 |
| Reference: | [2] Andradi H., Ho W. K.: On a new convergence class in sup-sober spaces.available at arXiv:1709.03269v1 [cs.LO] (2017), 13 pages. |
| Reference: | [3] Andradi H., Shen C., Ho W. K., Zhao D.: A new convergence inducing the $SI$-topology.Filomat 32 (2018), no. 17, 6017–6029. MR 3899335, 10.2298/FIL1817017A |
| Reference: | [4] Bandelt H.-J., Erné M.: The category of $Z$-continuous posets.J. Pure and Appl. Algebra 30 (1983), no. 3, 219–226. MR 0724033, 10.1016/0022-4049(83)90057-9 |
| Reference: | [5] Baranga A.: $Z$-continuous posets.Discrete Math. 152 (1996), no. 1–3, 33–45. MR 1388630, 10.1016/0012-365X(94)00307-5 |
| Reference: | [6] Engelking R.: General Topology.Mathematical Monographs, 60, PWN—Polish Scientific Publishers, Warszawa, 1977. Zbl 0684.54001, MR 0500780 |
| Reference: | [7] Erné M.: The ABC of order and topology.Category Theory at Work, Bremen, 1990, Res. Exp. Math., 18, Heldermann, Berlin, 1991, pages 57–83. MR 1147919 |
| Reference: | [8] Erné M.: $Z$-continuous posets and their topological manifestation.Applications of ordered sets in computer science, Braunschweig, 1996, Appl. Categ. Structures 7 (1999), no. 1–2, 31–70. MR 1714179, 10.1023/A:1008657800278 |
| Reference: | [9] Erné M.: Infinite distributive laws versus local connectedness and compactness properties.Topology Appl. 156 (2009), no. 12, 2054–2069. MR 2532134, 10.1016/j.topol.2009.03.029 |
| Reference: | [10] Gierz G., Hofmann K. H., Keimel K., Lawson J. D., Mislove M., Scott D. S.: Continuous Lattices and Domains.Encyclopedia of Mathematics and Its Applications, 93, Cambridge University Press, Cambridge, 2003. Zbl 1088.06001, MR 1975381 |
| Reference: | [11] Gierz G., Lawson J. D.: Generalized continuous and hypercontinuous lattices.Rocky Mountain J. Math. 11 (1981), no. 2, 271–296. MR 0619676, 10.1216/RMJ-1981-11-2-271 |
| Reference: | [12] Gierz G., Lawson J. D., Stralka A.: Quasicontinuous posets.Houston J. Math. 9 (1983), no. 2, 191–208. MR 0703268 |
| Reference: | [13] Goubault-Larrecq J.: Non-Hausdorff Topology and Domain Theory.New Mathematical Monographs, 22, Cambridge University Press, Cambridge, 2013. MR 3086734 |
| Reference: | [14] Heckmann R., Keimel K.: Quasicontinuous domains and the Smyth powerdomain.Proc. of the Twenty-Ninth Conf. Mathematical Foundations of Programming Semantics, MFPS XXIX, Electron. Notes Theor. Comput. Sci., 298, Elsevier, Amsterdam, 2013, pages 215–232. MR 3138523 |
| Reference: | [15] Kou H., Liu Y.-M., Luo M.-K.: On meet-continuous dcpos.Domain Theory, Logic and Computation, Semant. Struct. Comput., 3, Kluwer Acad. Publ., Dordrecht, 2003, pages 137–149. MR 2068007 |
| Reference: | [16] Lawson J. D.: $T_{0}$-spaces and pointwise convergence.Topology Appl. 21 (1985), no. 1, 73–76. MR 0808725, 10.1016/0166-8641(85)90059-8 |
| Reference: | [17] Lu J., Zhao B., Wang K.: $SI$-continuous spaces and continuous posets.Topology Appl. 264 (2019), 313–321. MR 3975753, 10.1016/j.topol.2019.06.032 |
| Reference: | [18] Mao X., Xu L.: Quasicontinuity of posets via Scott topology and sobrification.Order 23 (2006), no. 4, 359–369. MR 2309700, 10.1007/s11083-007-9054-4 |
| Reference: | [19] Mao X., Xu L.: Meet continuity properties of posets.Theoret. Comput. Sci. 410 (2009), no. 42, 4234–4240. MR 2561483, 10.1016/j.tcs.2009.06.017 |
| Reference: | [20] Venugopalan P.: Quasicontinuous posets.Semigroup Forum 41 (1990), no. 2, 193–200. MR 1057590, 10.1007/BF02573390 |
| Reference: | [21] Wright J. B., Wagner E. G., Thatcher J. W.: A uniform approach to inductive posets and inductive closure.Theoret. Comput. Sci. 7 (1978), no. 1, 57–77. MR 0480224, 10.1016/0304-3975(78)90040-3 |
| Reference: | [22] Zhao D., Ho W. K: On topologies defined by irreducible sets.J. Log. Algebr. Methods Program. 84 (2015), no. 1, 185–195. MR 3292951, 10.1016/j.jlamp.2014.10.003 |
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