Topologies on products of partially ordered sets I

Interval topologies

verfasst von: Marcel Erné
Abstract: Given a certain construction principle assigning to each partially ordered set P some topology θ(P) on P, one may ask under what circumstances the topology θ(P) of a product P = ⊗_j∈J P_j of partially ordered sets P_i agrees with the product topology ⊗_j∈Jθ(P_i) on P. We shall discuss this question for several types of interval topologies (Part I), for ideal topologies (Part II), and for order topologies (Part III). Some of the results contained in this first part are listed below: (1) Let θ_i(P) denote the segment topology. For any family of posets P_j ⊗_j∈Jθ_s(P_j)=θ_s(⊗_j∈JP_i) iff at most a finite number of the P_j has more than one element (1.1). (2) Let θ_cs(P) denote the co-segment topology (lower topology). For any family of lower directed posets P_j ⊗_j∈Jθ_cs(P_i)=θ_cs(⊗_j∈JP_i) iff each P_j has a least element (1.5). (3) Let θ_i(P) denote the interval topology. For a finite family of chains P_j, P_j ⊗_j∈Jθ_i(P_i)=θ_i(⊗_j∈JP_i) iff for all j∈k, P_j has a greatest element or P_k has a least element (2.11). (4) Let θ_ni(P) denote the new interval topology. For any family of posets P_j, P_j ⊗_j∈Jθ_ni(P_j)=θ_ni(⊗_j∈JP_j) whenever the product space is a b-space (i.e. a space where the closure of any subset Y is the union of all closures of bounded subsets of Y) (3.13). In the case of lattices, some of the results presented in this paper are well-known and have been shown earlier in the literature. However, the case of arbitrary posets often proved to be more difficult.
Organisationseinheit(en): Institut für Algebra, Zahlentheorie und Diskrete Mathematik
Typ: Artikel
Journal: Algebra universalis
Band: 11
Seiten: 295-311
Anzahl der Seiten: 17
ISSN: 0002-5240
Publikationsdatum: 12.1980
Publikationsstatus: Veröffentlicht
Peer-reviewed: Ja
ASJC Scopus Sachgebiete: Algebra und Zahlentheorie
Elektronische Version(en): https://doi.org/10.1007/BF02483109 (Zugang: Geschlossen)

BibTeX

@article{7973962919fb4ba3b4b13cfb13392645,
title = "Topologies on products of partially ordered sets I: Interval topologies",
abstract = "Given a certain construction principle assigning to each partially ordered set P some topology θ(P) on P, one may ask under what circumstances the topology θ(P) of a product P = ⊗j∈J P j of partially ordered sets P i agrees with the product topology ⊗j∈Jθ(P i) on P. We shall discuss this question for several types of interval topologies (Part I), for ideal topologies (Part II), and for order topologies (Part III). Some of the results contained in this first part are listed below: (1) Let θi(P) denote the segment topology. For any family of posets P j ⊗j∈Jθs(Pj)=θs(⊗j∈JPi) iff at most a finite number of the P j has more than one element (1.1). (2) Let θcs(P) denote the co-segment topology (lower topology). For any family of lower directed posets P j ⊗j∈Jθcs(Pi)=θcs(⊗j∈JPi) iff each P j has a least element (1.5). (3) Let θi(P) denote the interval topology. For a finite family of chains P j, P j ⊗j∈Jθi(Pi)=θi(⊗j∈JPi) iff for all j∈k, P j has a greatest element or P k has a least element (2.11). (4) Let θni(P) denote the new interval topology. For any family of posets P j, P j ⊗j∈Jθni(Pj)=θni(⊗j∈JPj) whenever the product space is a b-space (i.e. a space where the closure of any subset Y is the union of all closures of bounded subsets of Y) (3.13). In the case of lattices, some of the results presented in this paper are well-known and have been shown earlier in the literature. However, the case of arbitrary posets often proved to be more difficult.",
author = "Marcel Ern{\'e}",
year = "1980",
month = dec,
doi = "10.1007/BF02483109",
language = "English",
volume = "11",
pages = "295--311",
journal = "Algebra universalis",
issn = "0002-5240",
publisher = "Birkhauser Verlag Basel",
number = "1",
}