CONTINUITY OF UNIVERSALLY MEASURABLE HOMOMORPHISMS
Answering a longstanding problem originating in Christensen’s seminal work on Haar null sets [Math. Scand. 28 (1971), 124–128; Israel J. Math. 13 (1972), 255–260; Topology and Borel Structure. Descriptive Topology and Set Theory with Applications to Functional Analysis and Measure Theory, North-Holl...
| Main Author: | |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Cambridge University Press
2019-01-01
|
| Series: | Forum of Mathematics, Pi |
| Subjects: | |
| Online Access: | https://www.cambridge.org/core/product/identifier/S2050508619000052/type/journal_article |
| _version_ | 1811156253151330304 |
|---|---|
| author | CHRISTIAN ROSENDAL |
| author_facet | CHRISTIAN ROSENDAL |
| author_sort | CHRISTIAN ROSENDAL |
| collection | DOAJ |
| description | Answering a longstanding problem originating in Christensen’s seminal work on Haar null sets [Math. Scand. 28 (1971), 124–128; Israel J. Math. 13 (1972), 255–260; Topology and Borel Structure. Descriptive Topology and Set Theory with Applications to Functional Analysis and Measure Theory, North-Holland Mathematics Studies, 10 (Notas de Matematica, No. 51). (North-Holland Publishing Co., Amsterdam–London; American Elsevier Publishing Co., Inc., New York, 1974), iii+133 pp], we show that a universally measurable homomorphism between Polish groups is automatically continuous. Using our general analysis of continuity of group homomorphisms, this result is used to calibrate the strength of the existence of a discontinuous homomorphism between Polish groups. In particular, it is shown that, modulo $\text{ZF}+\text{DC}$, the existence of a discontinuous homomorphism between Polish groups implies that the Hamming graph on $\{0,1\}^{\mathbb{N}}$ has finite chromatic number. |
| first_indexed | 2024-04-10T04:48:32Z |
| format | Article |
| id | doaj.art-ac8cf65b46da4437ad4f025ba3915486 |
| institution | Directory Open Access Journal |
| issn | 2050-5086 |
| language | English |
| last_indexed | 2024-04-10T04:48:32Z |
| publishDate | 2019-01-01 |
| publisher | Cambridge University Press |
| record_format | Article |
| series | Forum of Mathematics, Pi |
| spelling | doaj.art-ac8cf65b46da4437ad4f025ba39154862023-03-09T12:34:26ZengCambridge University PressForum of Mathematics, Pi2050-50862019-01-01710.1017/fmp.2019.5CONTINUITY OF UNIVERSALLY MEASURABLE HOMOMORPHISMSCHRISTIAN ROSENDAL0Department of Mathematics, Statistics, and Computer Science (M/C 249), University of Illinois at Chicago, 851 S. Morgan St., Chicago, IL 60607-7045, USA;Answering a longstanding problem originating in Christensen’s seminal work on Haar null sets [Math. Scand. 28 (1971), 124–128; Israel J. Math. 13 (1972), 255–260; Topology and Borel Structure. Descriptive Topology and Set Theory with Applications to Functional Analysis and Measure Theory, North-Holland Mathematics Studies, 10 (Notas de Matematica, No. 51). (North-Holland Publishing Co., Amsterdam–London; American Elsevier Publishing Co., Inc., New York, 1974), iii+133 pp], we show that a universally measurable homomorphism between Polish groups is automatically continuous. Using our general analysis of continuity of group homomorphisms, this result is used to calibrate the strength of the existence of a discontinuous homomorphism between Polish groups. In particular, it is shown that, modulo $\text{ZF}+\text{DC}$, the existence of a discontinuous homomorphism between Polish groups implies that the Hamming graph on $\{0,1\}^{\mathbb{N}}$ has finite chromatic number.https://www.cambridge.org/core/product/identifier/S2050508619000052/type/journal_article03E15 (primary)22A0543A05 (secondary) |
| spellingShingle | CHRISTIAN ROSENDAL CONTINUITY OF UNIVERSALLY MEASURABLE HOMOMORPHISMS Forum of Mathematics, Pi 03E15 (primary) 22A05 43A05 (secondary) |
| title | CONTINUITY OF UNIVERSALLY MEASURABLE HOMOMORPHISMS |
| title_full | CONTINUITY OF UNIVERSALLY MEASURABLE HOMOMORPHISMS |
| title_fullStr | CONTINUITY OF UNIVERSALLY MEASURABLE HOMOMORPHISMS |
| title_full_unstemmed | CONTINUITY OF UNIVERSALLY MEASURABLE HOMOMORPHISMS |
| title_short | CONTINUITY OF UNIVERSALLY MEASURABLE HOMOMORPHISMS |
| title_sort | continuity of universally measurable homomorphisms |
| topic | 03E15 (primary) 22A05 43A05 (secondary) |
| url | https://www.cambridge.org/core/product/identifier/S2050508619000052/type/journal_article |
| work_keys_str_mv | AT christianrosendal continuityofuniversallymeasurablehomomorphisms |