ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Séria I: PRACE MATEMATYCZNE XXI (1979)
An d r z e j Gu t e k (Katowice)
On biconnected spaces without dispersion points
Abstract. Edwin W. Miller [11] has shown that if continuum hypothesis is true, then there exists a biconnected space, being a subspace of the Euclidean plane, which contains no dispersion point. This paper establishes that if hypothesis (Q) of Hausdorff and the strong Baire category theorem for reals, known consequences of Martin’s axiom, are true, then such a biconnected space exists.
Introduction. B. Knaster and C. Kuratowski initiated in [6] studies of biconnected spaces, i.e. such connected spaces which cannot be decomposed into two non-intersecting non-degenerate connected subspaces. Each of bi
connected spaces constructed by Knaster and Kuratowski contains a dispers
ion point; a point p of connected space X is said to be a dispersion point if X — [p] contains no non-degenerate connected subspace. Kuratowski [8]
raised the question whether every biconnected space contains sucfi a point.
Kline has proved in [5] that every biconnected space contains at most one dispersion point. Wilder [16] has shown that a biconnected space X which contains no dispersion point cannot contain finite dispersion sets (a set A is a dispersion set of space X if X —A contains no non-degenerate connected subspace). Biconnected spaces having dispersion points were extensively studied by R. Duda in [2].
P. M. Swingle introduced [15] widely connected spaces, i.e. connected spaces such that every connected non-degenerate subspace of which is dense.
He constructed such spaces and proved that if X is a biconnected and dense subspace of an indecomposable continuum, then X is widely connected.
It is known that widely connected spaces have no cut points, in particular they have no dispersion points. Miller proved in [11] that a widely connected subspace X of an indecomposable continuum in the Euclidean plane is biconnected (and has no dispersion point) if the family of the so-called
^-boundaries does not possess property B, where an X-boundary is the common part of X and of a plane continuum which separates the plane between some points of X . A family of sets is said to possess property В if there exists a set which contains at least one element of each set of the
family and does not exhaust any set of the family. Miller constructed such biconnected space using continuum hypothesis.
This paper establishes that to perform Miller’s construction it suffices to assume the Hausdorff hypothesis (Q) and the strong Baire category theorem for reals.
In what follows со will denote the set of positive integers as well as the first infinite ordinal. Cardinal numbers will be regarded as initial ordinal numbers.
1. Set theoretical preliminaries. Define quasi-order on subsets of со by letting А с В if A — В is finite and B — A is infinite. Let (Q) be the proposition :
(Q) Let В — {B„: песо} and A = [Aa: a < m}, where m < 2W, be families of infinite subsets of со. If
A , a
* A. B„ B- S i,
then there is a subset D of со such that Ая <= D c= Bn for every et < m and песо.
A special case of (Q) when the family A has cardinality аох was investigated by Hausdorff in [4]. For that special case proposition (Q) was studied by Rothberger, [12] and [13].
Paper [3] by Engelking contains a survey of results concerning (Q).
The negation of that special case of (Q) is called there H 1.
Let (Q') be the proposition:
(Q') Let В = {Bn: песо} and A = {Ая: a < m}, where m < 201, be families of infinite subsets o f со. I f ... c Bn c= ... <== B2 <= Вг and Ая c= Bn for every песо and <x < m, then there is a subset D of со such that Aa cz D cz B„
for every песо and a < m.
Lem m a 1. (Q) holds if and only if (Q').
Clearly, (Q') => (Q). We shall prove that (Q) implies (Q').
Let m be the least cardinal (less than 2е0) for which (Q') fails. Then there are families A = [A7: a < m\ and В = {Bn: песо} of infinite subsets of со such that ... <= Bn cz ... c= B2 B l and Aa a Bn for every a < m and песо, and there is no subset D of со such that Ax cz D <= Bn for every a < m and песо.
We shall define by induction the family A' = {Af. a < m} such that Ая с: Л' and Ai с А'л cz Bn for < a < m and песо.
Let A \ = A x.
Suppose that A^, where ^ < a, such that Д* c A^ and A’n cz A'^ <= Bn for г/ < Ç < a and песо are already defined. Since a < m, (Q') can be applied to the family {Af. Ç < a), i.e. there is a subset Dx of со such that A'i <= A< ^ for ^ < a and песо. Let A'a = 4 u D r There is Ai а Ая a Bn for £ < a and песо, and Ая c A f
The family A' = {Af. a < m} just defined is such that
A'l <=: *2 a: b„ B: cz B 1 * 1
By (Q), there is a subset D of со such that A} cz D c z Bn for every a < m and песо; this contradics the definition of m.
Lemma 2. Assume (Q). If X is a dense in itself Hausdorff space with a denumerable basis T, and A and В are disjoint subsets of X , card {A) is less than 2W, and В is denumerable, then there exists a subset V of T such that
V x = {t e T ' : x e t } is finite if x e A , and infinite if x eB .
P roof. Let
T = {t„: песо},
A = [ал: a < m}, where m = card(,4) < 20), В ={/)„: nèco},
Ал = {iEco: а, фи].
Let us associate with each point bn of В a set i{n) = {i1 (n), i2 (n) , ...} such that f(n) < i2(n) < and tij + l(n) cz tij(n) and such that [bn] = Ç) {tij(n)\ jEco}.
We put Bn = (J {i(k) : к ^ n}. Clearly, since X is a dense in itself Hausdorff space, the sets А л and Bn, a < m and п е с о, are infinite and В i cz B2 cz ... cz Bn cz ... and Bn cz Ax. Let us observe that if A' cz B', then (со —В') cz (со —A'). Thus, letting A'x = со — A., and B'n = со —Bn, we have
. . . cz B’n cz . . . cz B2 cz B\ and A'x cz B’n for a < m and п е с о. By Lemma 1 and (Q') there is a subset D ' of со such that Л* cz D ' c z B'n for every
a < m and п е с о. Passing to the complements and letting D = co — D ' , we have Bn cz D cz Ал.
Let V = {t„6T: h eD). There is
card V x = card {tne T' : x E t n} = card [h eD: x E t n}, card V ал = card {h eD: a x E t n} = card (D — Ax), if axEA, card V b n = card {/cgD: b „ E t k} = card (D n B„), if Ь„еВ.
We see that T 'x is finite if x eA , and infinite if x eB.
Lemma 3. Assume (Q). I f X is a dense in itself Hausdorff space with a denumerable basis T, and A and В are disjoint subsets of X , card (A) is less than 2m, and В is denumerable and dense in X , then there exists a denumerable collection F of nowhere dense closed subsets of X such that A cz [j F and B n ( j F = 0.
P roof. Let T = {t„: песо}. Since card (A) is less than 2W, by Lemma 2 there exists a subset T of T such that V x = {t„ET': x E t n} is finite if
x eA and infinite if x eB. Let W„ = [J {t kE V : к > n}. Since В cz Wn for each песо, so each Wnis dense and open in X. There is A n f) {Wn: песо}
~ 0. Passing to the complements, we infer that the sets Fn = X — Wn are nowhere dense and closed, A cz (J [Fn: песо} and B n ( j {F„: песо} = 0.
- Prace Matematyczne 21.1
Recall that a set A is meager if there is a denumerable collection F of nowhere dense closed sets such that A <= (J F.
The strong Baire category theorem for reals is the proposition (cf.
Martin and Solovay [10], p. 170)
(SBCT) The union of m meager subset of reals is meager for every m < 2(,}.
Both propositions, (Q) and (SBCT), are known to be consequences of Martin’s axiom, and even of hypothesis (P) which is a consequence of Martin’s axiom (Booth [1]).
Hypothesis (P) asserts the following:
(P) I f A i 3 A 2 ... £ < m, and m < 2°\ is a sequence of infinite subsets of со, then there is an infinite subset В of m such that В cz A* for c < m.
Rothberger [13] proved that (P) implies (Q) in the case where the family of sets A* is of the cardinality m = The proof for other m, m < 2W, can be found in paper by Kucia and Szymanski [7]. Implication (P) => (SBCT) is easy, being analogous to the proof of the usual Baire Theorem.
Assumptions (Q) and (SBCT) together with the negation of continuum hypothesis are consistent with the theory ZFC, since, as was proved by Solovay and Tennenbaum in [14], Martin’s axiom and the negation of continuum hypothesis are consistent with ZFC.
2. A lemma on C x i . Let / be the unit closed segment [0, 1] of reals, and let C be the Cantor set in /. Let Pc and Pt be the projections of C x i onto C and I, respectively.
A selection of a map p: X Y is a map / : Y -» X such that p(f(y))
= y for y e Y.
Le m m a 4. Assume (Q). Let A and В be disjoint subsets of C x i such that card A is less than 2(0 and В is denumerable and dense. Then there is a continuous selection / : C C x i of the projection Pc such that the image f ( C) has no point in common with A and such that the common part of f ( C) and В is dense in f{C).
P ro o f. Without the loss of generality we may assume that C is given by the usual ternary construction, i.e. that
С = С ^ С г и . . . , and C„ — (J {C" : q ... in is a dyadic sequence),
1 where C" f are closed-open disjoint intervals of C,
C"- i l 0 u C j i 1 = C”t_ 1 i and diam Cit f
ll " ‘n - l u * 1 — *fi — 1 A 'l - ' n - l ' l - ' n -5И
We shall consider two cases.
C a se 1. We assume that P[{A) and P[(B) are disjoint subsets of I.
By Lemma 3, there is a denumerable collection F = {F„: n e w } of nowhere dense closed subsets of / such that {JF contains P, (A) and has no point in common with P;(B).
An 1/'-neighbourhood U of point x e / is an open subinterval U of I such that x e U and such that diam U < l/i.
Take points a0 and in В such that a0 eCo><I and а хе С } х / (this is possible because В is dense in C x i ) . The projections PI (a0) and Р Д а / do not belong to F Thus, there exist 1-neighbourhoods U0 and L/ of
P, ( a / such that cl((7o)n ^ i = 0 and cl ( Ul) r \ F l = 0.
Suppose that for each к, к ^ n, there are defined points af of В and 1/k-neighbourhoods t/fx...*fe of Pi (u^ ...;/ such that
(1) с1(Сг1..л ) п Р к = 0,
(2) <=
(3) ûi1...ike(Cf1...ifcx UiV"ik), (4) if Р с К . . . < , . , ) € С ? , . . . then « к ,..* .*
Now, for each C7,+ 1i we let ait f ; , , = a ; if РЛя,-, ,• )e G C"1+.1.i„ift+1 • In the other case we take as а11...,п + 1 any point of В n (C".+ 1 i ; x l / | ; ). We construct 1
p i ( a iv ..in + 1) so that
cl ( U i 1 . . . in + l)r^Fn + l = 0 and
^ neighbourhoods Uil__in + 1 of
U,.
This is possible, since F , ( B ) n f . t l = 0 and +1 e B n (C?*.1 y „ +1 x
Thus, the points ail...i„ + 1 and sets having properties (1)—(4) are defined.
Let S„ = (J {Cniv in x cl UiV"in: / ... t„ is (0 ,1 } -sequence}. Let S = П {S„:
n ew }. Define a function / : C - > C x I by setting { f ( y) } = { y } x I n S ; the last set consists of (exactly) one point since diam Ut t < — . Clearly the
1 n
function / is continuous and is the desired selection of Pc . The set / (C) of values of / is disjoint with A (property (1)), and B n f ( C ) is dense in /(C ) (property (3)).
C ase 2. When Pt {A) and Pi{B) intersect, we reduce the construction to the preceding case as follows. We consider C x i as a. subset of the plane.
There are less than 2ю pairs (p, q), where p e A and q e B , and therefore less than 2ю directions of segments pq joining points in these pairs. Thus there is a dense set of directions, with cardinality 2“, such that no line having such a direction has points in common with both sets A and B. Take
a direction from that set such that the projection R in that direction of the point (1, 1) of C x i onto the line containing the segment {0} x I lies between the points (0,0) and (0, 1). So, the sets P(/4) and R( B) are disjoint. Consider the parallelogram P contained in I x l between lines x = 0, x = 1, and lines having the above chosen direction going through points (0, 0) and (1, 1).
The set P n C x i is homeomorphic to C x i , the set В r \ P is denumer
able and dense in P n C x i , card (A n P) is less than 2W, and R (A n P) and R(B n P) are disjoint subsets of the segment R(P). Applying the argument from Case 1 to these sets and to the projections Pc\pncxi and R instead of the rectangular projections Pc and P/? we get the desired selection.
It is easy to see that by simple modifications of the proof we can obtain two selections f x and f 2 of Pc such that 0 < Р,(Л(Х)) < P ; ( /2(x)) < 1 for x e C .
These two selections, restricted to a closed-open interval C of C such that 0, 1 $ C , define a closed simple curve J in Int I x l such that J n C x i is the union of graphs of these selections.
3. Application to the construction of a biconnected set without dispersion point.
Th eo r e m. Assume (Q) and (SBCT) (or (P)). The Euclidean plane contains a biconnected set with no dispersion point.
P ro o f. Let К be an indecomposable plane continuum such that the % common part of К and of the unit square I x l is C x i (a known Knaster continuum (cf. [9], p. 204) can be regarded as such a continuum). Let D be a denumerable and dense subset of C x i . Let {Bx: a < 2W} be the family of plane continua which separate K. Let [Dx: a < 2W] be the collection of subsets of D dense in some open subset of C x i . Let {Са: a < 2ю} be the collection of composants of K.
Miller [11] has shown that if we add to the set D points px from Bx n K if Вх сл D = 0, in such a way that
(i) for а Ф f , the points px and рр lie in different composants of К (note that pa are defined only in the case where Bx n D = 0), and such that
(ii) for each a < 2W there exists a simple closed curve Jx separating К which does not contain points pp, < 2°\ and such that Jx n D a Dx, '
then the resulting set M is a connected subset of К and has no connected non-degenerate subsets in composants. These properties imply (in view of Theorem 7 from [11]) that M is widely connected. On the other hand, the family {Bx: a < 2<0] has not property В (taking a selection only from curves Jx, one of them is exhausted by the values of that selection (see Miller’s proof in [11])). Thus (in view of Theorem 6 from [11]), the set M, as a widely connected subset of an indecomposable continuum, is biconnected and has no dispersion point.
The aim of our proof is to show that the construction can be performed under assumptions (Q) and (SBCT).
We sketch the construction describing in details only the claims which require modifications in comparison with Miller’s construction.
We shall construct in fact for each a a simple closed curve Ja having much more special form, namely
(iii) the common part of Jx and C x i is a union of graphs of two disjoint selections of the projection Pc defined on the same closed-open subset of C,
(iv) Jx lies in I x l and Jx n D is dense in Jx n C x I .
C o n str u c tio n . Suppose that px and Jx, satisfying (i)-(iv), for a < ft are already constructed. Consider Bp.
If Bp r \ D Ф 0, we have only to define Jp. Let W be an open subset of C x i such that РС(Ж) is a closed-open subset of C, 0,1 f P c (W), W — PC{W) x P} (W), and in which the set Dp is dense. Applying Lemma 4 to the sets W, Dp , and to the set consisting of px, a < /?, and of points of D — Dp, we obtain two selections f x and f 2 of the projection Pc such that Pi ( h (x)) < P j ( f 2 (x)) for x e P c (W). These selections define the simple closed curve Jp so that conditions (ii)-(iv) are satisfied for a ^ f (condition (i) is satisfied vacuously).
If Bp c\ D — 0, we proceed as follows.
Let E be the set of those composants which do not intersect the set consisting of points px, a < fj, points lying in C x I over the end points of complementary intervals of C, and which do not contain points of D.
Let us observe that there are less than 2W composants which do not belong to E. So, the set C x {0} n (J E is of the second category in C x (0).
If Bp intersects 2W composants outside of Int l x I, then we take as pp any point of Bp lying outside of Int l x l and belonging to a composant of E.
If not, then Bp intersects 2W composants in Int / х / . Therefore, there are 2W composants of E which intersect Bp in Int I x I. Let us associate with each such a composant Сц a non-degenerate subcontinuum Ku of Bp lying entirely in Int I x I. If each Rц is a segment of a line of C x i , then we take as pp any point of which is not a point of Ja, a < j8 (it is possible since each J%, a < /?, intersects Rfl in at most two points and
card ft < 2W). If not, then there is a subcontinuum Rv of Bp r \ I x l which intersects two different composants. So, Pc (Rv n C x I) is a non-empty interval of C. The projection of Jx r\ Rv n C x I onto C is closed and nowhere dense in Pc (Rv n C x I ) . So, by (SBCT), the set Pc ([j {Rv n C x I n j a: a < ft}) is of the first category in Pc (Rv n C x I). Then there is a point y of PC(R v n C x I) which belongs to a composant from the set E. The set {y} x I lies in the composant chosen above. We take pp as a point of { y } x I n R v. Condition (i) is clearly satisfied. Now, we apply Lemma 4 to obtain the curve Jp such that (ii)-(iv) hold for a ^ ft, in the similar way as before.
This finishes the induction.
References
[1] D. D. B oo th , Countably indexed ultrafilters, Thesis, University of Wisconsin, Madison 1969.
[2] R. D u d a, On biconnected sets with dispersion points, Rozprawy Mat. 37 (1964).
[3] R. E n g e lk in g , Hausdorff's gaps and limits and compactifications, in: Theory of sets and topology, VEB Deutscher Verlag der Wissenschaften, Berlin 1972, p. 89-93.
[4] F. H a u sd o rff, Summen von X^Mengen, Fund. Math. 26 (1936), p. 241-255.
[5] J. R. K line, A theorem concerning connected point sets, ibidem 3 (1922), p. 238-239.
[6] B. K n a s te r and K. K u ra to w s k i, Sur les ensemble connexes, ibidem 2 (1921), p. 206-255.
[7] A. K u c ia and A. S zy m a n sk i, There exist absolute points which are not minimal, Colloq. Math. 37 (1977), p. 29-34.
[8] K. K u ra to w s k i, Problème 19, Fund. Math. 3 (1922), p. 322.
[9] — Topology, vol. II, Academic Press, New York and London 1968.
[10] D. A. M a rtin and R. M. S o lo v ay , Internal Cohen extensions, Ann. Math. Logic 2 (1970), p. 143-178.
[11] E. W. M iller, Concerning biconnected sets, Fund. Math. 29 (1937), p. 123-133.
[12] F. R o th b e rg e r, Sur un ensemble toujours de première catégorie qui est dépourvu de la propriété X, ibidem 32 (1939), p. 294-300.
[13] — Sur les familles indénombrables de suites de nombres naturelles et les problèmes concer
nant la propriété C, Proc. Cambridge Phil. Soc. 37 (1941), p. 109-126.
[14] R. M. S o lo v ay and S. T e n n e n b a u m , Iterated Cohen extensions and Souslin’s problem, Ann. of Math. 94 (1971), p. 201-245.
[15] P. M. S w ingle, Two types of connected sets, Bull. Amer. Math. Soc. 37 (1931), p. 254-258.
[16] R. L. W ild er, On the dispersion sets of connected point sets, Fund. Math. 6 (1925), p. 214-228.
