summary:A Mazurkiewicz set $M$ is a subset of a plane with the property that each straight line intersects $M$ in exactly two points. We modify the original construction to obtain a Mazurkiewicz set which does not contain vertices of an equilateral triangle or a square. This answers some questions by L.D. Loveland and S.M. Loveland. We also use similar methods to construct a bounded noncompact, nonconnected generalized Mazurkiewicz set

Charatonik, Marta N.

Charatonik, Włodzimierz J.

A Mazurkiewicz set M is a subset of a plane with the property that each straight line intersects M in exactly two points. We modify the original construction to obtain a Mazurkiewicz set which does not contain vertices of an equilateral triangle or a square. This answers some questions by L.D. Loveland and S.M. Loveland. We also use similar methods to construct a bounded noncompact, nonconnected generalized Mazurkiewicz set

Charatonik, W. J.

Missouri University of Science and Technology (Missouri S&T): Scholars' Mine

On Mazurkiewicz Sets

Czech digital mathematics library

On Mazurkiewicz sets

Institute of Mathematics AS CR, v. v. i.

Commentationes Mathematicae Universitatis CarolinaeMarta N. Charatonik; Włodzimierz J. CharatonikOn Mazurkiewicz setsCommentationes Mathematicae Universitatis Carolinae, Vol. 41 (2000), No. 4, 817--819Persistent URL: http://dml.cz/dmlcz/119213Terms of use:© Charles University in Prague, Faculty of Mathematics and Physics, 2000Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document mustcontain these Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech DigitalMathematics Library http://project.dml.czComment.Math.Univ.Carolin. 41,4 (2000)817–819 817On Mazurkiewicz setsMarta N. Charatonik, W lodzimierz J. CharatonikAbstract. A Mazurkiewicz set M is a subset of a plane with the property that eachstraight line intersects M in exactly two points. We modify the original constructionto obtain a Mazurkiewicz set which does not contain vertices of an equilateral triangleor a square. This answers some questions by L.D. Loveland and S.M. Loveland. Wealso use similar methods to construct a bounded noncompact, nonconnected generalizedMazurkiewicz set.Keywords: Mazurkiewicz set, GM-set, double midset propertyClassification: Primary 54C99, 54F15, 54G20; Secondary 54B20By a Mazurkiewicz set (shortly M-set) we mean a subset X of the plane suchthat every straight line intersects X in exactly two points. It was constructedin [3] using transfinite induction. The notion was generalized in two directions:to generalized Mazurkiewicz sets (GM-sets) and to sets with the double midsetproperty (DMP-sets). Let us recall that a subset X of the plane is a GM-set if itcontains at least two points and each line that separates two points of X intersectsX in exactly two points. A subset X of the plane is a DMP-set if it contains atleast two points and the perpendicular bisector of every segment joining two pointsin X intersect X in exactly two points. It follows from the definitions that everyM-set is a GM-set and every GM-set is an DMP-set. For more information aboutthese notions see [2]. In the same article the authors ask some questions relatedto the subject. Here we answer some of them in a more general case constructing,using transfinite induction, an M-set with some additional geometrical properties.Namely, the M-set that does not contain vertices of an equilateral triangle orvertices of a square, and whose image under the inversion with respect to the unitcircle is a bounded, noncompact, nonconnected GM-set.We will need some denotation. The symbol c denotes the cardinal numbercontinuum, i.e. the first ordinal number whose cardinality is the cardinality ofreals. All the constructions are going to be done in the complex plane C. Givenx, y ∈ C the symbol l(x, y) denotes the line through x and y if x 6= y, andl(x, x) = {x}. If both x and y are distinct from 0, and x 6= y, then c(x, y) isthe circle that contains x, y and 0. Moreover we put c(x, x) = {x}. For a subsetA ⊂ C we put L(A) =⋃{l(x, y) : x, y ∈ A} and C(A) =⋃{c(x, y) : x, y ∈ A}.We denote by B the open unit disk in the plane, i.e. B = {z ∈ C : |z| < 1}.818 M.N.Charatonik, W.J.CharatonikTheorem. There is an M-set A satisfying the following conditions:(1) A does not contain vertices of an equilateral triangle;(2) A does not contain vertices of a right isosceles triangle;(3) A ∩ cl B = ∅;(4) any circle that contains 0 and is not contained in cl B intersects A atexactly two points.Proof: Given two different points a, b ∈ C define P (a, b) as the set of allpoints x ∈ C such that the triangle with vertices a, b, x is an equilateral one or aright isosceles one. Thus P (a, b) has exactly eight points. In particular we haveP (0, 1) = {i,−i, 1+ i, 1− i, 12 +√22 i,12 +−√22 i,12 +√32 i,12 +−√32 i}. Additionallywe put P (x, x) = ∅. For a set A ⊂ C let P (A) =⋃{P (x, y) : x, y ∈ A}.Let {lα : α ≤ c} be the set of all straight lines in the plane, and let {cα : α ≤ c}be the set of all circles passing through 0 and not contained in cl B. We willdefine, for α < c, the set Aα, and A =⋃{Aα : α < c} will be the required M-set.Assume that, for some α < c, the sets Aβ for β < α, have been defined satisfyingthe following conditions:(1β) card(Aβ) < c;(2β) for every γ < β we have Aγ ⊂ Aβ ;(3β) Aβ ∩ lβ is a two point set;(4β) Aβ ∩ cβ is a two point set;(5β) Aβ ∩ P (Aβ) = ∅;(6β) Aβ contains no three colinear points;(7β) there is no circle in the plane that contains three different points of Aβand the point 0;(8β) Aβ ∩ cl B = ∅.Put Nα =⋃{Aβ : β < α}. Then• card(P (Nα)) < c,• card(lα ∩ (⋃{l(x, y) : x, y ∈ Nα})) < c,• card(cα ∩ (⋃{c(x, y) : x, y ∈ Nα})) < c,• card(lα ∩ Nα) ≤ 2,• card(cα ∩ Nα) ≤ 2.Thus we can choose points xα, yα, zα, tα that satisfy the following conditions,where Gα = cl B ∪ P (Nα) ∪ L(Nα) ∪ C(Nα).• xα, yα ∈ lα \ cα,• zα, tα ∈ cα \ lα,• if card(lα ∩ Nα) = 2, then {xα, yα} = lα ∩ Nα,• if card(cα ∩ Nα) = 2, then {zα, tα} = cα ∩ Nα,• if card(lα ∩ Nα) = 1, then {xα} = lα ∩ Nα and yα /∈ Gα,• if card(cα ∩ Nα) = 1, then {zα} = cα ∩ Nα and tα /∈ Gα,• if card(lα ∩ Nα) = 0, then xα, yα /∈ Gα,• if card(cα ∩ Nα) = 0, then zα, tα /∈ Gα.On Mazurkiewicz sets 819Finally put Aα = Nα ∪ {xα, yα, zα, tα}. One can verify that, by the construc-tion, conditions (1α)–(8α) are satisfied. Putting A =⋃{Aα : α < c} we see thatA is the required M-set. This finishes the proof. Remark 1. In [2, Questions 2 and 3, p. 488] the authors asked if there is aDMP-set in the plane that does not contain vertices of a square (Question 2) andif there is a DMP-set in the plane that does not contain vertices of an equilateraltriangle (Question 3). Because every M-set is a DMP-set, the Theorem answersboth questions.Denote by h : C \ {0} → C \ {0} the inversion with respect to the unit circle,i.e. h(z) = 1/z. Observe that h(h(z)) = z.Proposition. Let A be an M-set that satisfies conditions (3) and (4) of theTheorem. Then h(A) is a GM-set.Proof: First observe that h(A) ⊂ B by condition (3). Let l be a line thatseparates two points of h(A). If 0 ∈ l, then h(l \ {0}) = l \ {0}. If 0 /∈ l, thenh(l) is a circle passing through 0 and not contained in B. In any case h(l) ∩ A isa two point set by (4), and therefore h(h(l))∩ h(A) = l ∩ h(A) is a two point set,as required. Remark 2. In [2, Question 6, p. 490] the authors ask the following question. Isthere a bounded GM-set which is not a simple closed curve? Is a bounded GM-set necessarily closed? Connected? Since the constructed set h(A) is a boundedGM-set homeomorphic to an M-set, it is neither closed (M-sets are not bounded,so not compact) nor connected (M-sets are zerodimensional, see [1, Theorem 2,p. 553]). Thus the Proposition answers in the negative all of the three questions.It also answers more particular Question 7 and partially Question 8.Acknowledgment. The authors would like to thank Janusz J. Charatonik forhis help in the preparation of this paper.References[1] Kulesza J., A two-point set must be zerodimensional, Proc. Amer. Math. Soc. 116 (1992),551–553.[2] Loveland L.D., Loveland S.M., Planar sets that line hits twice, Houston J. Math. 23 (1997),485–497.[3] Mazurkiewicz S., Sur un ensemble plan qui a avec chaque droite deux et seulement deuxpoints communs, C.R. Soc. de Varsovie 7 (1914), 382–383.W.J. Charatonik:Mathematical Institute, University of Wroc law, pl. Grunwaldzki 2/4,50-384 Wroc law, PolandandDepartment of Mathematics and Statistics, University of Missouri-Rolla, Rolla,MO 65409, USAE-mail : wjcharat@math.uni.wroc.plwjcharat@umr.edu(Received January 10, 2000)

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On Mazurkiewicz sets

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