Cardinal invariants of monotone and porous sets. (English) Zbl 1245.28003

Authors’ abstract: A metric space \((X,d)\) is monotone if there is a linear order \(<\) on \(X\) and a constant \(c\) such that \(d(x,y)\leqslant c d(x,z)\) for all \(x<y<z\) in \(X\). We investigate cardinal invariants of the \(\sigma \)-ideal Mon generated by monotone subsets of the plane. Since there is a strong connection between monotone sets in the plane and porous subsets of the line, plane and the Cantor set, cardinal invariants of these ideals are also investigated. In particular, we show that non(Mon)\(\geqslant \mathfrak {m}_{\sigma -\text{linked}}\), but non(Mon)\(<\mathfrak {m}_{\sigma -\text{centered}}\) is consistent. Also cov(Mon)\(< \mathfrak {c}\) and cof\((\mathcal N) <\) cov(Mon) are consistent.


28A75 Length, area, volume, other geometric measure theory
03E15 Descriptive set theory
03E17 Cardinal characteristics of the continuum
03E35 Consistency and independence results
54H05 Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets)
Full Text: DOI Euclid


[1] Tomek Bartoszyński and Haim Judah Set theory , A K Peters Ltd., Wellesley, MA,1995.
[2] Andreas Blass and Saharon Shelah There may be simple \(P_{\aleph_1}\) - and \(P_{\aleph_2}\)-points and the Rudin-Keisler ordering may be downward directed, Annals of Pure and Applied Logic , vol. 33(1987), no. 3, pp. 213-243. · Zbl 0634.03047
[3] Jörg Brendle The additivity of porosity ideals , Proceedings of the American Mathematical Society , vol. 124(1996), no. 1, pp. 285-290. · Zbl 0839.03029
[4] Ilijas Farah OCA and towers in \({\mathcal P}(\text{\bfseries\upshape N})/{\mathrm fin}\), Commentationes Mathematicae Universitatis Carolinae , vol. 37(1996), no. 4, pp. 861-866.
[5] E. Järvenpää, M. Järvenpää, A. Käenmäki, and V. Suomala Asympotically sharp dimension estimates for \(k\) -porous sets, Mathematica Scandinavica , vol. 97(2005), no. 2, pp. 309-318. · Zbl 1097.28005
[6] Kenneth Kunen Set theory , Studies in Logic and the Foundations of Mathematics, vol. 102, North-Holland, Amsterdam,1980.
[7] Pertti Mattila Distribution of sets and measures along planes , Journal of the London Mathematical Society, Second Series , vol. 38(1988), no. 1, pp. 125-132. · Zbl 0618.28005
[8] —- Geometry of sets and measures in Euclidean spaces , Cambridge Studies in Advanced Mathematics, vol. 44, Cambridge University Press, Cambridge,1995.
[9] Aleš Nekvinda and Ondřej Zindulka A Cantor set in the plane that is not \(\sigma\)-monotone , Fundamenta Mathematicae , vol. 213(2011), no. 3, pp. 221-232. · Zbl 1227.54037
[10] —- Monotone metric spaces , Order , · Zbl 1260.54045
[11] Miroslav Repický Porous sets and additivity of Lebesgue measure , Real Analysis Exchange , vol. 15(1989/90), no. 1, pp. 282-298. · Zbl 0716.28004
[12] —- Additivity of porous sets , Real Analysis Exchange , vol. 16(1990/91), no. 1, pp. 340-343.
[13] —- Cardinal invariants related to porous sets , Set theory of the reals \((\)Ramat Gan, 1991\()\), Israel mathematics conference proceedings , vol. 6, Bar-Ilan University, Ramat Gan,1993, pp. 433-438.
[14] Diego Rojas-Rebolledo Using determinacy to inscribe compact non-\(\sigma\) -porous sets into non-\(\sigma\)-porous projective sets, Real Analysis Exchange , vol. 32(2006/07), no. 1, pp. 55-66. · Zbl 1120.03027
[15] Arto Salli On the Minkowski dimension of strongly porous fractal sets in \(\text{\bfseries\upshape R}^n\) , Proceedings of the London Mathematical Society. Third Series , vol. 62(1991), no. 2, pp. 353-372. · Zbl 0716.28006
[16] L. Zajíček Porosity and \(\sigma\) -porosity, Real Analysis Exchange , vol. 13(1987/88), no. 2, pp. 314-350. · Zbl 0666.26003
[17] —- On \(\sigma\) -porous sets in abstract spaces, Abstract and Applied Analysis ,(2005), no. 5, pp. 509-534. · Zbl 1098.28003
[18] Luděk Zajíček and Miroslav Zelený Inscribing closed non-\(\sigma\) -lower porous sets into Suslin non-\(\sigma\)-lower porous sets, Abstract and Applied Analysis ,(2005), no. 3, pp. 221-227. · Zbl 1091.28001
[19] Jindřich Zapletal Forcing idealized , Cambridge Tracts in Mathematics, vol. 174, Cambridge University Press, Cambridge,2008.
[20] —- Preserving \(P\) -points in definable forcing, Fundamenta Mathematicae , vol. 204(2009), no. 2, pp. 145-154. · Zbl 1174.03022
[21] Miroslav Zelený An absolutely continuous function with non-\(\sigma\) -porous graph, Real Analysis Exchange , vol. 30(2004/05), no. 2, pp. 547-563. · Zbl 1108.28004
[22] Miroslav Zelený and Luděk Zajíček Inscribing compact non-\(\sigma\) -porous sets into analytic non-\(\sigma\)-porous sets, Fundamenta Mathematicae , vol. 185(2005), no. 1, pp. 19-39. · Zbl 1086.54025
[23] Ondřej Zindulka Universal measure zero, large Hausdorff dimension, and nearly Lipschitz maps , Fundamenta Mathematicae , submitted.
[24] —- Mapping Borel sets onto balls by Lipschitz and nearly Lipschitz maps , in preparation.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.