Paper Info
Title
Finite Intervals in the Lattice of Topologies
Abstract
We discuss the question whether every finite interval in the lattice of all topologies on some set is isomorphic to an interval in the lattice of all topologies on a finite set – or, equivalently, whether the finite intervals in lattices of topologies are, up to isomorphism, exactly the duals of finite intervals in lattices of quasiorders. The answer to this question is in the affirmative at least for finite atomistic lattices. Applying recent results about intervals in lattices of quasiorders, we see that, for example, the five-element modular but non-distributive lattice cannot be an interval in the lattice of topologies. We show that a finite lattice whose greatest element is the join of two atoms is an interval of T0-topologies iff it is the four-element Boolean lattice or the five-element non-modular lattice. But only the first of these two selfdual lattices is an interval of orders because order intervals are known to be dually locally distributive.
Year
DOI
Venue
2000
10.1023/A:1008647423701
Applied Categorical Structures
Keywords
Field
DocType
t0-topology.,mathematics subject classifications 2000: 06b15,interval,54f05. key words: atomistic,lattice of topologies,54d35,distributive lattice
Topology,Discrete mathematics,Congruence lattice problem,Combinatorics,Lattice model (physics),Comparison of topologies,Distributive lattice,Complemented lattice,Map of lattices,Lattice problem,Integer lattice,Mathematics
Journal
Volume
Issue
ISSN
8
1-2
1572-9095
Citations 
PageRank 
References 
1
0.63
0
Authors
1
Name
Order
Citations
PageRank
Jürgen Reinhold1315.89