| Title:
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The dual group of a dense subgroup (English) |
| Author:
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Comfort, W. W. |
| Author:
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Raczkowski, S. U. |
| Author:
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Trigos-Arrieta, F. Javier |
| Language:
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English |
| Journal:
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Czechoslovak Mathematical Journal |
| ISSN:
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0011-4642 (print) |
| ISSN:
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1572-9141 (online) |
| Volume:
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54 |
| Issue:
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2 |
| Year:
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2004 |
| Pages:
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509-533 |
| Summary lang:
|
English |
| . |
| Category:
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math |
| . |
| Summary:
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Throughout this abstract, $G$ is a topological Abelian group and $\widehat{G}$ is the space of continuous homomorphisms from $G$ into the circle group $\mathbb{T}$ in the compact-open topology. A dense subgroup $D$ of $G$ is said to determine $G$ if the (necessarily continuous) surjective isomorphism $\widehat{G}\twoheadrightarrow \widehat{D}$ given by $h\mapsto h\big |D$ is a homeomorphism, and $G$ is determined if each dense subgroup of $G$ determines $G$. The principal result in this area, obtained independently by L. Außenhofer and M. J. Chasco, is the following: Every metrizable group is determined. The authors offer several related results, including these. 1. There are (many) nonmetrizable, noncompact, determined groups. 2. If the dense subgroup $D_i$ determines $G_i$ with $G_i$ compact, then $\oplus _iD_i$ determines $\Pi _i G_i$. In particular, if each $G_i$ is compact then $\oplus _i G_i$ determines $\Pi _i G_i$. 3. Let $G$ be a locally bounded group and let $G^+$ denote $G$ with its Bohr topology. Then $G$ is determined if and only if ${G^+}$ is determined. 4. Let $\mathop {\mathrm non}({\mathcal N})$ be the least cardinal $\kappa $ such that some $X \subseteq {\mathbb{T}}$ of cardinality $\kappa $ has positive outer measure. No compact $G$ with $w(G)\ge \mathop {\mathrm non}({\mathcal N})$ is determined; thus if $\mathop {\mathrm non}({\mathcal N})=\aleph _1$ (in particular if CH holds), an infinite compact group $G$ is determined if and only if $w(G)=\omega $. Question. Is there in ZFC a cardinal $\kappa $ such that a compact group $G$ is determined if and only if $w(G)<\kappa $? Is $\kappa =\mathop {\mathrm non}({\mathcal N})$? $\kappa =\aleph _1$? (English) |
| Keyword:
|
Bohr compactification |
| Keyword:
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Bohr topology |
| Keyword:
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character |
| Keyword:
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character group |
| Keyword:
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Außenhofer-Chasco Theorem |
| Keyword:
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compact-open topology |
| Keyword:
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dense subgroup |
| Keyword:
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determined group |
| Keyword:
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duality |
| Keyword:
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metrizable group |
| Keyword:
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reflexive group |
| Keyword:
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reflective group |
| MSC:
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03E35 |
| MSC:
|
03E50 |
| MSC:
|
22A05 |
| MSC:
|
22A10 |
| MSC:
|
22B99 |
| MSC:
|
22C05 |
| MSC:
|
43A40 |
| MSC:
|
54D30 |
| MSC:
|
54E35 |
| MSC:
|
54H11 |
| idZBL:
|
Zbl 1080.22500 |
| idMR:
|
MR2059270 |
| . |
| Date available:
|
2009-09-24T11:15:01Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/127907 |
| . |
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