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Keywords:
two point set; partial two point set; complete nonmeasurability; Hamel basis; Marczewski measurable set; Marczewski null; $s$-nonmeasurability; Luzin set; Sierpiński set
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).
References:
[1] Bell, J. L., Slomson, A. B.: Models and Ultraproducts. An Introduction. North-Holland Publishing Company, Amsterdam (1969). MR 0269486 | Zbl 0179.31402
[2] Carlson, T. J.: Strong measure zero and strongly meager sets. Proc. Am. Math. Soc. 118 (1993), 577-586. DOI 10.1090/S0002-9939-1993-1139474-6 | MR 1139474 | Zbl 0787.03037
[3] Cichoń, J., Morayne, M., Rałowski, R., Ryll-Nardzewski, C., Żeberski, S.: On nonmeasurable unions. Topology Appl. 154 (2007), 884-893. DOI 10.1016/j.topol.2006.09.013 | MR 2294636 | Zbl 1109.03049
[4] Dijkstra, J. J., Kunen, K., Mill, J. van: Hausdorff measures and two point set extensions. Fundam. Math. 157 (1998), 43-60. MR 1623614
[5] Dijkstra, J. J., Mill, J. van: Two point set extensions---a counterexample. Proc. Am. Math. Soc. 125 (1997), 2501-2502. DOI 10.1090/S0002-9939-97-03875-6 | MR 1396973
[6] Jech, T.: Set Theory. The third millennium edition, revised and expanded. Springer Monographs in Mathematics Springer, Berlin (2003). MR 1940513 | Zbl 1007.03002
[7] Kraszewski, J., Rałowski, R., Szczepaniak, P., Żeberski, S.: Bernstein sets and $\kappa$-coverings. Math. Log. Q. 56 (2010), 216-224. DOI 10.1002/malq.200910008 | MR 2650240
[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). MR 0597342 | Zbl 0443.03021
[9] Larman, D. D.: A problem of incidence. J. Lond. Math. Soc. 43 (1968), 407-409. DOI 10.1112/jlms/s1-43.1.407 | MR 0231724 | Zbl 0157.53702
[10] Mauldin, R. D.: On sets which meet each line in exactly two points. Bull. Lond. Math. Soc. 30 (1998), 397-403. DOI 10.1112/S0024609397004268 | MR 1620829 | Zbl 0931.28001
[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 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. <a href="http://mathscinet.ams.org/mathscinet-getitem?mr=0250248" target="_blank">MR 0250248</a></div> <div class="reference">[12] Miller, A. W.: <b>Infinite combinatorics and definability</b>. Ann. Pure Appl. Logic 41 (1989), 179-203. <a href="http://dx.doi.org/10.1016/0168-0072(89)90013-4" target="_blank">DOI 10.1016/0168-0072(89)90013-4</a> | <a href="http://mathscinet.ams.org/mathscinet-getitem?mr=0983001" target="_blank">MR 0983001</a> | <a href="http://www.zentralblatt-math.org/zmath/en/search/?q=an:0667.03037" target="_blank">Zbl 0667.03037</a></div> <div class="reference">[13] Miller, A. W., Popvassiliev, S. G.: <b>Vitali sets and Hamel base that are Marczewski measurable</b>. Fundam. Math. 166 (2000), 269-279. <a href="http://mathscinet.ams.org/mathscinet-getitem?mr=1809419" target="_blank">MR 1809419</a></div> <div class="reference">[14] Rałowski, R.: <b>Remarks on nonmeasurable unions of big point families</b>. Math. Log. Q. 55 (2009), 659-665. <a href="http://dx.doi.org/10.1002/malq.200810014" target="_blank">DOI 10.1002/malq.200810014</a> | <a href="http://mathscinet.ams.org/mathscinet-getitem?mr=2582166" target="_blank">MR 2582166</a> | <a href="http://www.zentralblatt-math.org/zmath/en/search/?q=an:1192.03025" target="_blank">Zbl 1192.03025</a></div> <div class="reference">[15] Rałowski, R., Żeberski, S.: <b>Completely nonmeasurable unions</b>. Cent. Eur. J. Math. 8 (2010), 683-687. <a href="http://dx.doi.org/10.2478/s11533-010-0038-z" target="_blank">DOI 10.2478/s11533-010-0038-z</a> | <a href="http://mathscinet.ams.org/mathscinet-getitem?mr=2671219" target="_blank">MR 2671219</a> | <a href="http://www.zentralblatt-math.org/zmath/en/search/?q=an:1207.03056" target="_blank">Zbl 1207.03056</a></div> <div class="reference">[16] Schmerl, J. H.: <b>Some 2-point sets</b>. Fundam. Math. 208 (2010), 87-91. <a href="http://dx.doi.org/10.4064/fm208-1-6" target="_blank">DOI 10.4064/fm208-1-6</a> | <a href="http://mathscinet.ams.org/mathscinet-getitem?mr=2609222" target="_blank">MR 2609222</a> | <a href="http://www.zentralblatt-math.org/zmath/en/search/?q=an:1196.03057" target="_blank">Zbl 1196.03057</a></div> <div class="reference">[17] Szpilrajn, E.: <b>Sur une classe de fonctions de M. Sierpiński et la classe correspondante d'ensembles</b>. Fundam. Math. 24 (1934), 17-34 French. <a href="https://doi.org/10.4064/fm-24-1-17-34" target="_blank">DOI 10.4064/fm-24-1-17-34</a></div> <div class="reference">[18] Żeberski, S.: <b>On completely nonmeasurable unions</b>. Math. Log. Q. 53 (2007), 38-42. <a href="http://dx.doi.org/10.1002/malq.200610024" target="_blank">DOI 10.1002/malq.200610024</a> | <a href="http://mathscinet.ams.org/mathscinet-getitem?mr=2288888" target="_blank">MR 2288888</a> | <a href="http://www.zentralblatt-math.org/zmath/en/search/?q=an:1109.03046" target="_blank">Zbl 1109.03046</a></div> </div> </div> </div> </div> <div id="ds-options"> <div id="search-box"> <h3>Search</h3> <div class="ds-option-set" id="ds-search-option"> <form method="get" id="ds-search-form" action="/manakin/search"> <fieldset> <input type="text" class="ds-text-field " name="query" /> <input value="Go" type="submit" name="submit" class="ds-button-field " onclick=" var radio = document.getElementById("ds-search-form-scope-container"); if (radio != undefined && radio.checked) { var form = document.getElementById("ds-search-form"); form.action= "/manakin/handle/" + radio.value + "/search" ; } " /> <label> <input checked="checked" value="" name="scope" type="radio" id="ds-search-form-scope-all" />Search</label> <br /> <label> <input name="scope" type="radio" id="ds-search-form-scope-container" value="10338.dmlcz/143603" />This Collection</label> </fieldset> </form> <a href="/manakin/advanced-search">Advanced Search</a> </div> </div> <div xmlns="http://www.w3.org/1999/xhtml" id="artifactbrowser_Navigation_list_browse" class="ds-option-set"> <ul class="ds-options-list"> <li> <h4 class="ds-sublist-head">Browse</h4> <ul class="ds-simple-list"> <li> <a href="/manakin/community-list">Collections</a> </li> <li> <a href="/manakin/browse-title">Titles</a> </li> <li> <a href="/manakin/browse-author">Authors</a> </li> <li> <a href="/manakin/MSCSubjects">MSC</a> </li> </ul> </li> </ul> </div> <div xmlns="http://di.tamu.edu/DRI/1.0/" class="ds-option-set" id="artifactbrowser_Navigation_list_account"> <h3 class="ds-option-set-head"> </h3> <a href="/about">About DML-CZ</a> </div> <div id="eudml-partner"> <div class="eudml-partner-head">Partner of</div> <a href="http://eudml.org/"> <img alt="EuDML logo" src="/manakin/themes/DML/eudml-logo-mensi.png" /> </a> </div> </div> <div id="ds-footer"> <div id="ds-footer-links">© 2010 <a href="http://www.math.cas.cz/">Institute of Mathematics CAS</a> </div> <div> <a class="mc-open-dialog" href="#">Cookies</a> </div> </div> </div> <script> !function(i,c){i.muniCookies=c;var s=document.createElement("script");s.src=c.scriptUrl+"main.js",document.head.appendChild(s)}(window,{ scriptUrl: 'https://cdn.muni.cz/Scripts/libs/muni-cookies/', lang: 'en', customStyle: true, key: 'b973f3ab-96c9-49e0-a330-761aba5a344d', colors: { primary: '#000000', primaryText: '#fff', link: '#336699' }}) </script> </body> </html>