TY - JOUR

T1 - Topologies on products of partially ordered sets I

T2 - Interval topologies

AU - Erné, Marcel

PY - 1980/12

Y1 - 1980/12

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=51649159993&partnerID=8YFLogxK

U2 - 10.1007/BF02483109

DO - 10.1007/BF02483109

M3 - Article

AN - SCOPUS:51649159993

VL - 11

SP - 295

EP - 311

JO - Algebra universalis

JF - Algebra universalis

SN - 0002-5240

IS - 1

ER -