| Title:
|
Two point sets with additional properties (English) |
| Author:
|
Bienias, Marek |
| Author:
|
Głąb, Szymon |
| Author:
|
Rałowski, Robert |
| Author:
|
Żeberski, Szymon |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
63 |
| Issue:
|
4 |
| Year:
|
2013 |
| Pages:
|
1019-1037 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
A subset of the plane is called a two point set if it intersects any line in exactly two points. We give constructions of two point sets possessing some additional properties. Among these properties we consider: being a Hamel base, belonging to some $\sigma $-ideal, being (completely) nonmeasurable with respect to different $\sigma $-ideals, being a $\kappa $-covering. We also give examples of properties that are not satisfied by any two point set: being Luzin, Sierpiński and Bernstein set. We also consider natural generalizations of two point sets, namely: partial two point sets and $n$ point sets for $n=3,4,\ldots , \aleph _0,$ $\aleph _1.$ We obtain consistent results connecting partial two point sets and some combinatorial properties (e.g. being an m.a.d. family). (English) |
| Keyword:
|
two point set |
| Keyword:
|
partial two point set |
| Keyword:
|
complete nonmeasurability |
| Keyword:
|
Hamel basis |
| Keyword:
|
Marczewski measurable set |
| Keyword:
|
Marczewski null |
| Keyword:
|
$s$-nonmeasurability |
| Keyword:
|
Luzin set |
| Keyword:
|
Sierpiński set |
| MSC:
|
03E35 |
| MSC:
|
03E75 |
| MSC:
|
15A03 |
| MSC:
|
28A05 |
| idZBL:
|
Zbl 06373959 |
| idMR:
|
MR3165512 |
| DOI:
|
10.1007/s10587-013-0069-2 |
| . |
| Date available:
|
2014-01-28T14:14:31Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143614 |
| . |
| Reference:
|
[1] Bell, J. L., Slomson, A. B.: Models and Ultraproducts. An Introduction.North-Holland Publishing Company, Amsterdam (1969). Zbl 0179.31402, MR 0269486 |
| Reference:
|
[2] Carlson, T. J.: Strong measure zero and strongly meager sets.Proc. Am. Math. Soc. 118 (1993), 577-586. Zbl 0787.03037, MR 1139474, 10.1090/S0002-9939-1993-1139474-6 |
| Reference:
|
[3] Cichoń, J., Morayne, M., Rałowski, R., Ryll-Nardzewski, C., Żeberski, S.: On nonmeasurable unions.Topology Appl. 154 (2007), 884-893. Zbl 1109.03049, MR 2294636, 10.1016/j.topol.2006.09.013 |
| Reference:
|
[4] Dijkstra, J. J., Kunen, K., Mill, J. van: Hausdorff measures and two point set extensions.Fundam. Math. 157 (1998), 43-60. MR 1623614 |
| Reference:
|
[5] Dijkstra, J. J., Mill, J. van: Two point set extensions---a counterexample.Proc. Am. Math. Soc. 125 (1997), 2501-2502. MR 1396973, 10.1090/S0002-9939-97-03875-6 |
| Reference:
|
[6] Jech, T.: Set Theory. The third millennium edition, revised and expanded.Springer Monographs in Mathematics Springer, Berlin (2003). Zbl 1007.03002, MR 1940513 |
| Reference:
|
[7] Kraszewski, J., Rałowski, R., Szczepaniak, P., Żeberski, S.: Bernstein sets and $\kappa$-coverings.Math. Log. Q. 56 (2010), 216-224. MR 2650240, 10.1002/malq.200910008 |
| Reference:
|
[8] Kunen, K.: Set Theory. An Introduction to Independence Proofs.Studies in Logic and the Foundations of Mathematics vol. 102 North-Holland Publishing Company, Amsterdam (1980). Zbl 0443.03021, MR 0597342 |
| Reference:
|
[9] Larman, D. D.: A problem of incidence.J. Lond. Math. Soc. 43 (1968), 407-409. Zbl 0157.53702, MR 0231724, 10.1112/jlms/s1-43.1.407 |
| Reference:
|
[10] Mauldin, R. D.: On sets which meet each line in exactly two points.Bull. Lond. Math. Soc. 30 (1998), 397-403. Zbl 0931.28001, MR 1620829, 10.1112/S0024609397004268 |
| Reference:
|
[11] Mazurkiewicz, S.: O pewnej mnogości płaskiej, która ma z każdą prostą dwa i tylko dwa punkty wspólne.Polish Comptes Rendus des Séances de la Société des Sciences et Lettres de Varsovie 7 (1914), 382-384 French transl <title>Sur un ensemble plan qui a avec chaque droite deux et seulement deux points communs Stefan Mazurkiewicz, Traveaux de Topologie et ses Applications K. Borsuk et al. Wydawnictwo naukowe PWN, Warsaw, 1969, 46-47. MR 0250248 |
| Reference:
|
[12] Miller, A. W.: Infinite combinatorics and definability.Ann. Pure Appl. Logic 41 (1989), 179-203. Zbl 0667.03037, MR 0983001, 10.1016/0168-0072(89)90013-4 |
| Reference:
|
[13] Miller, A. W., Popvassiliev, S. G.: Vitali sets and Hamel base that are Marczewski measurable.Fundam. Math. 166 (2000), 269-279. MR 1809419 |
| Reference:
|
[14] Rałowski, R.: Remarks on nonmeasurable unions of big point families.Math. Log. Q. 55 (2009), 659-665. Zbl 1192.03025, MR 2582166, 10.1002/malq.200810014 |
| Reference:
|
[15] Rałowski, R., Żeberski, S.: Completely nonmeasurable unions.Cent. Eur. J. Math. 8 (2010), 683-687. Zbl 1207.03056, MR 2671219, 10.2478/s11533-010-0038-z |
| Reference:
|
[16] Schmerl, J. H.: Some 2-point sets.Fundam. Math. 208 (2010), 87-91. Zbl 1196.03057, MR 2609222, 10.4064/fm208-1-6 |
| Reference:
|
[17] Szpilrajn, E.: Sur une classe de fonctions de M. Sierpiński et la classe correspondante d'ensembles.Fundam. Math. 24 (1934), 17-34 French. 10.4064/fm-24-1-17-34 |
| Reference:
|
[18] Żeberski, S.: On completely nonmeasurable unions.Math. Log. Q. 53 (2007), 38-42. Zbl 1109.03046, MR 2288888, 10.1002/malq.200610024 |
| . |
