Finite Intervals in the Lattice of Topologies

Jürgen Reinhold ()
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Jürgen Reinhold: Universität Hannover, Institut für Mathematik

A chapter in Papers in Honour of Bernhard Banaschewski, 2000, pp 367-376 from Springer

Abstract: 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 T 0-topologies if 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.

Keywords: atomistic; interval; lattice of topologies; T 0-topology.; 06B15; 54D35; 54F05 (search for similar items in EconPapers)
Date: 2000
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-94-017-2529-3_22

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DOI: 10.1007/978-94-017-2529-3_22

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