Locally Quasi-Convex Compatible Topologies on a Topological Group
For a locally quasi-convex topological abelian group (G,τ), we study the poset \(\mathscr{C}(G,τ)\) of all locally quasi-convex topologies on (G) that are compatible with (τ) (i.e., have the same dual as (G,τ) ordered by inclusion. Obviously, this poset has always a bottom element, namely the weak t...
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MDPI AG
2015-10-01
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| Series: | Axioms |
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| Online Access: | http://www.mdpi.com/2075-1680/4/4/436 |
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| author | Lydia Außenhofer Dikran Dikranjan Elena Martín-Peinador |
| author_facet | Lydia Außenhofer Dikran Dikranjan Elena Martín-Peinador |
| author_sort | Lydia Außenhofer |
| collection | DOAJ |
| description | For a locally quasi-convex topological abelian group (G,τ), we study the poset \(\mathscr{C}(G,τ)\) of all locally quasi-convex topologies on (G) that are compatible with (τ) (i.e., have the same dual as (G,τ) ordered by inclusion. Obviously, this poset has always a bottom element, namely the weak topology σ(G,\(\widehat{G})\) . Whether it has also a top element is an open question. We study both quantitative aspects of this poset (its size) and its qualitative aspects, e.g., its chains and anti-chains. Since we are mostly interested in estimates ``from below'', our strategy consists of finding appropriate subgroups (H) of (G) that are easier to handle and show that \(\mathscr{C} (H)\) and \(\mathscr{C} (G/H)\) are large and embed, as a poset, in \(\mathscr{C}(G,τ)\). Important special results are: (i) if \(K\) is a compact subgroup of a locally quasi-convex group \(G\), then \(\mathscr{C}(G)\) and \(\mathscr{C}(G/K)\) are quasi-isomorphic (3.15); (ii) if (D) is a discrete abelian group of infinite rank, then \(\mathscr{C}(D)\) is quasi-isomorphic to the poset \(\mathfrak{F}_D\) of filters on D (4.5). Combining both results, we prove that for an LCA (locally compact abelian) group \(G \) with an open subgroup of infinite co-rank (this class includes, among others, all non-σ-compact LCA groups), the poset \( \mathscr{C} (G) \) is as big as the underlying topological structure of (G,τ) (and set theory) allows. For a metrizable connected compact group \(X\), the group of null sequences \(G=c_0(X)\) with the topology of uniform convergence is studied. We prove that \(\mathscr{C}(G)\) is quasi-isomorphic to \(\mathscr{P}(\mathbb{R})\) (6.9). |
| first_indexed | 2024-04-13T02:14:10Z |
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| id | doaj.art-d7429b0e919341bc8b0ee0847e5843b5 |
| institution | Directory Open Access Journal |
| issn | 2075-1680 |
| language | English |
| last_indexed | 2024-04-13T02:14:10Z |
| publishDate | 2015-10-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj.art-d7429b0e919341bc8b0ee0847e5843b52022-12-22T03:07:12ZengMDPI AGAxioms2075-16802015-10-014443645810.3390/axioms4040436axioms4040436Locally Quasi-Convex Compatible Topologies on a Topological GroupLydia Außenhofer0Dikran Dikranjan1Elena Martín-Peinador2Faculty of Computer Science and Mathematics, Universität Passau, Innstr. 33, Passau D-94032, GermanyDepartment of Mathematics and Computer Science, University of Udine, Via delle Scienze, 208-Loc. Rizzi, Udine 33100, ItalyInstituto de Matemática Interdisciplinar y Departamento de Geometría y Topología, Universidad Complutense de Madrid, Madrid 28040, SpainFor a locally quasi-convex topological abelian group (G,τ), we study the poset \(\mathscr{C}(G,τ)\) of all locally quasi-convex topologies on (G) that are compatible with (τ) (i.e., have the same dual as (G,τ) ordered by inclusion. Obviously, this poset has always a bottom element, namely the weak topology σ(G,\(\widehat{G})\) . Whether it has also a top element is an open question. We study both quantitative aspects of this poset (its size) and its qualitative aspects, e.g., its chains and anti-chains. Since we are mostly interested in estimates ``from below'', our strategy consists of finding appropriate subgroups (H) of (G) that are easier to handle and show that \(\mathscr{C} (H)\) and \(\mathscr{C} (G/H)\) are large and embed, as a poset, in \(\mathscr{C}(G,τ)\). Important special results are: (i) if \(K\) is a compact subgroup of a locally quasi-convex group \(G\), then \(\mathscr{C}(G)\) and \(\mathscr{C}(G/K)\) are quasi-isomorphic (3.15); (ii) if (D) is a discrete abelian group of infinite rank, then \(\mathscr{C}(D)\) is quasi-isomorphic to the poset \(\mathfrak{F}_D\) of filters on D (4.5). Combining both results, we prove that for an LCA (locally compact abelian) group \(G \) with an open subgroup of infinite co-rank (this class includes, among others, all non-σ-compact LCA groups), the poset \( \mathscr{C} (G) \) is as big as the underlying topological structure of (G,τ) (and set theory) allows. For a metrizable connected compact group \(X\), the group of null sequences \(G=c_0(X)\) with the topology of uniform convergence is studied. We prove that \(\mathscr{C}(G)\) is quasi-isomorphic to \(\mathscr{P}(\mathbb{R})\) (6.9).http://www.mdpi.com/2075-1680/4/4/436locally quasi-convex topologycompatible topologyquasi-convex sequencequasi-isomorphic posetsfree filtersMackey groups |
| spellingShingle | Lydia Außenhofer Dikran Dikranjan Elena Martín-Peinador Locally Quasi-Convex Compatible Topologies on a Topological Group Axioms locally quasi-convex topology compatible topology quasi-convex sequence quasi-isomorphic posets free filters Mackey groups |
| title | Locally Quasi-Convex Compatible Topologies on a Topological Group |
| title_full | Locally Quasi-Convex Compatible Topologies on a Topological Group |
| title_fullStr | Locally Quasi-Convex Compatible Topologies on a Topological Group |
| title_full_unstemmed | Locally Quasi-Convex Compatible Topologies on a Topological Group |
| title_short | Locally Quasi-Convex Compatible Topologies on a Topological Group |
| title_sort | locally quasi convex compatible topologies on a topological group |
| topic | locally quasi-convex topology compatible topology quasi-convex sequence quasi-isomorphic posets free filters Mackey groups |
| url | http://www.mdpi.com/2075-1680/4/4/436 |
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