Details
| Original language | English |
|---|---|
| Article number | 106981 |
| Journal | Topology and its applications |
| Volume | 273 |
| Publication status | Published - 15 Mar 2020 |
Abstract
We study general notions of convergence and continuity in arbitrary spaces or ordered sets, extending considerably topological concepts in domain theory like those of Scott convergence, alias lower (lim-inf) convergence, and Scott topology. It turns out that the convergence-theoretical properties of being localized, a limit relation, pretopological, or topological, respectively, all correspond to important properties of the underlying ordered sets that reduce to (meet) continuity and similar properties in the classical situation. Basic tools are the cut closure operators and diverse order-theoretical or topological variants of them. We characterize the generalized Scott convergence spaces abstractly as so-called core determined convergence spaces. This unifying concept provides simplifications and new insights into various areas of order theory, topology and theoretical computer science. In particular, some intimate connections between convergence properties, meet preservation by certain closure operations, and the continuity of meet operations are established.
Keywords
- Closure, Continuous, Convergence, Cut, Ideal, Meet, Nucleus, Scott topology, Web space
ASJC Scopus subject areas
- Mathematics(all)
- Geometry and Topology
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In: Topology and its applications, Vol. 273, 106981, 15.03.2020.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Generalized continuous closure spaces I
T2 - Meet preserving closure operations
AU - Erné, Marcel
PY - 2020/3/15
Y1 - 2020/3/15
N2 - We study general notions of convergence and continuity in arbitrary spaces or ordered sets, extending considerably topological concepts in domain theory like those of Scott convergence, alias lower (lim-inf) convergence, and Scott topology. It turns out that the convergence-theoretical properties of being localized, a limit relation, pretopological, or topological, respectively, all correspond to important properties of the underlying ordered sets that reduce to (meet) continuity and similar properties in the classical situation. Basic tools are the cut closure operators and diverse order-theoretical or topological variants of them. We characterize the generalized Scott convergence spaces abstractly as so-called core determined convergence spaces. This unifying concept provides simplifications and new insights into various areas of order theory, topology and theoretical computer science. In particular, some intimate connections between convergence properties, meet preservation by certain closure operations, and the continuity of meet operations are established.
AB - We study general notions of convergence and continuity in arbitrary spaces or ordered sets, extending considerably topological concepts in domain theory like those of Scott convergence, alias lower (lim-inf) convergence, and Scott topology. It turns out that the convergence-theoretical properties of being localized, a limit relation, pretopological, or topological, respectively, all correspond to important properties of the underlying ordered sets that reduce to (meet) continuity and similar properties in the classical situation. Basic tools are the cut closure operators and diverse order-theoretical or topological variants of them. We characterize the generalized Scott convergence spaces abstractly as so-called core determined convergence spaces. This unifying concept provides simplifications and new insights into various areas of order theory, topology and theoretical computer science. In particular, some intimate connections between convergence properties, meet preservation by certain closure operations, and the continuity of meet operations are established.
KW - Closure
KW - Continuous
KW - Convergence
KW - Cut
KW - Ideal
KW - Meet
KW - Nucleus
KW - Scott topology
KW - Web space
UR - http://www.scopus.com/inward/record.url?scp=85077141533&partnerID=8YFLogxK
U2 - 10.1016/j.topol.2019.106981
DO - 10.1016/j.topol.2019.106981
M3 - Article
AN - SCOPUS:85077141533
VL - 273
JO - Topology and its applications
JF - Topology and its applications
SN - 0166-8641
M1 - 106981
ER -