Pacific Journal of Mathematics

A WILD CANTOR SET IN THE HILBERT CUBE

RAYMOND Y. T. WONG

Vol. 24, No. 1 May 1968 PACIFIC JOURNAL OF MATHEMATICS Vol. 24, No. 1, 1968

A WILD CANTOR SET IN THE HILBERT CUBE

RAYMOND Y. T. WONG

Let En be the Euclidean w-space. A Cantor set C is a set homeomorphic with the Cantor middle-third set. Antoine and Blankinship have shown that there exists a "wild" Cantor set in any En for n ^ 3, where "wild" means that En - C is not simply connected. However it is also known that no "wild" Cantor set (in fact, compact set) can exist in many infinite dimensional spaces, such as s (the countably infinite product of

lines) or the Hubert space l2. A result of this paper provides a positive answer for a generalization of Blankinship's result in the Hubert cube.

If X is a space, we denote by Xn the space Π?=3 X% and X°° the space ΠΓ=i X% with X{ = X Let τn denote the projecting function n of X°° onto X and πn the projecting function of X°° onto Xn. Let J, J denote intervals [— 1,1], (— 1,1) respectively. The Hubert cube is the space J°° under the metric p(x, y) = Σ;>i(l χι ~ Vi I)/2*- Hubert space, l2, is the space of all square summable sequences of real numbers with metric d((Xi), (#<)) = τ/[ΣΓ=i(^ - Vi)2]- The space JΓ°° is also denoted by s. Let E% = ΠΓ=i Ei be the Euclidean π-space. A Cantor set is a set homeomorphic with the Cantor middle-third set. The existence of a Cantor set C in E* (n 5> 3) such that En - C is not simply connected was first demonstrated by Antoine [4] in 1921 and constructed by W. A. Blankinship [5] in 1951. It is known that every Cantor set is s (or in l2) must be tame, in the sense that its complement in s (or in l2) is topologically as nics as the space itself. In fact it has been proved (by V. Klee in the case of l2 [9] and by R. D. Anderson [1] in the case of s, using Klee's method) that if if is a compact set in X (for X — s or ϊ2), then X — K ^ X. The question as to whether a finite dimensional closed set can leave the Hubert cube multiply connected (in particular, whether a Cantor set can have this property) was then raised in [5] by Blankinship and was also later mentioned in [7] by Klee. In this paper we shall give such a question a positive answer by constructing a Cantor set C in the Hubert cube J°° such that J°° — C is not homotopically trivial. In fact, we shall apply the result of Blankinship [5] to show that J°° — C has nontrivial lst-Homotopy group. We remark that such a set C cannot be constructed as a subset of j°°. Note that Anderson [1] (by using Klee's method) proved that any Cantor set C (in fact, any compact set) in j00 can be carried into an end-face, say

Kγ — {x e J°° I πx(x) = 1}, by a homeomorphism on J°°. It is quite clear that the complement of any Cantor subset (in fact, any compact subset)

189 190 RAYMOND Y. T. WONG of Ki in J°° is homotopically trivial, therefore, if the complement of C in J°° is to be homotopically nontrivial, C must, in a sense, join various end-faces of J°°.

2* Some notation and lemma* All homeomorphisms concerned are assumed to be geometric homeomorphisms, and when a home- omorphism has domain in En, it is assumed to be linear. Two subsets of En are similar if they are homeomorphic under some homeomorphism. Let Δ denote the boundary of the unit square in E2. A *-circle is a set homeomorphic to A. An %-tube, n ^ 3, is a set homeomorphic to the product of a circular 2-cell with (n — 2) *-circles.

We shall choose a fixed set of positive real numbers ru r2, with the properties that (1) n > 1 and (2) rn+1 > 2(ΣLin). Let L{ =

[r, = 1, r, + 1] c Et and 2/ = Π?=i U x K+i, rΛ+2> •)• We shall regard £7" as a subset of £fw+1 by considering En as £^0.

LEMMA 1. If X is a Hausdorff space and Au A2, is a decreas- ing sequence of compact subsets of X such that each Aι is dense in itself, then UΓ=i^» ^s dense in itself.

Proof. If x is an isolated point of ΠΓ=i^i, then for some i, x is an isolated point of Aif contrary to the hypothesis.

3* Brief outline of the construction* The construction is an inductive modification of the construction by Antoine [4] and by Blankinship [5]. The Cantor set C will be the intersection of a decreasing sequence of compact subsets Ku K2, of the Hubert cube

L°° = ΠΓ=i£; For each n ^ 3, Kn will be the product of a compact n subset K'n of L with ΠΓ=»+i Li- Kl is the intersection of a simple chain of linking 3-tubes of E5 with ZΛ K{ will be contained in Kζ x 4 L4 and is the intersection of a simple chain of linking 4-tubes of E with L4 and so on.

4* Construction of if3*

DEFINITION. Let r, s be positive integers and dr an arbitrary real number. Let S be a compact subset of £roo(= Π" ι ^) such that

πr(S) = dr. We say S is the set generated by rotating S about the hyperplane xr = dr and xs = 0 if

« _ (a; G #- : 3?/ G S 3 (xr, x8) e Bd([dr - y8, dr + ys] x [ - y8, ys]) ( and Xi = yi for i Φ r, s

where [dr - yt, dr + ?/sl c Er,[- y8, ys] c E8. A WILD CANTOR SET IN THE HILBERT CUBE 191

The following Lemma is evident:

LEMMA 2. Suppose S is the set defined above and πs(S) > 0, then S is homeomorphίc to the product of S with a *-circle.

DEFINITION. Let

2 2 2 T = {x e E~ : (x, - n) + (x2 - r2) ^(—Y and xt = r< for i ^ 3}

2 2 Λ = {x e E- : (x, - n) + (x* ~ r2) = (~γj and x{ = r< for i ^ 3} .

For n ^ 3, define T% inductively to be the set generated by rotating n ι T ~ about the hyperplane ajΛ_1 = 0, a;n = rn.

% LEMMA 3. For n :> 2, min τr%(Γ ) ^ 1.

Proof. It is clear for % = 2. For n Ξ> 3, it follows from the n fact min πn(T ) = rn — (1/4 + r2 + + rw_!) and from the hypothesis of r{.

w n LEMMA 4. For n ^ 3, T is αw n-tube in E .

2 Proof. τr2(T ) > 0 by Lemma 3. Then by Lemma 2, Γ is a 3- tube. Inductively, Tn is an n-tuhe.

n n 2 LEMMA 5. For n^3,T ΠL = τ2(T ) x Π?=s U x (rn+1, rn+2f ...).

Proof. This is a consequence of Lemma 3.

Let {ίj}|=i be a chain of cylically linked disjoint 3-tubes contained in the interior of Γ3 and looping once around the axis of T3. We assume (1) they are all similar to T3, (2) I = 0 (mod 4) and I is large enough so that each t\ can be regarded as the set generated by ro- tating a small circular 2-cell t\ along a small *-circle 4i9 (3) diam(^) < l/3(diam T3) for all i, and (4) Only two members of {£3}Li intersect Bd(L3)(one in each side) and the intersection of each such t\ with Bd B 3 (K ) is exactly two disjoint 2-cells. Let A3 = U<=i *<, Ki = 43 ίl L and

κ3 = Kζ x πr=4 Lim

5. Construction of K4, K6, ••• . For the purpose of simplicity, we shall give only the construction of iΓ4 and assert that for n ^ 5,

Kn can be inductively constructed.

Step 1. For each ί, let hi be a (linear) homeomorphism of T3 192 RAYMOND Y. T. WONG

onto t\. Hence {ί?y = /^(ίj)}^ is a similar chain of cyclically linked disjoint 3-tubes in t\. We require that each hi is so chosen that (1) if t\ is a member that intersects Bd(L3), then only two members of ι 3 {tlj} j==1 intersect Bd(L ) and the intersection of each such member with Bd(L3) is exactly two disjoint 2-cells and (2) diam^ ) < (l/32)diam(T3) for all ij.

4 Step 2. For each i,j, let t\5 be the 4-tube in T generated by rotating i\ά about planes x3 = 0, x4 = r4. We now regard each t\ό as the set generated by rotating a small 2-cell t\5 along a small *-circle. 3 We assume further that t\5 is contained in L whenever ί^ intersects

ZΛ Let ??,- be the set generated by rotating t\ά about planes x2 = 0,

£* = n. Then ί^ can be regarded as the geometric product of t\ά 3 with Ai5. tlj is a 3-tube. Let hi5 be a linear homeomorphism of T onto ??i# Let t\jk = hi5(tk), k = 1, 2, , I. We require each hi5 is so 2 3 ι chosen that (1) if t iS c L , then only two members of {fijk) k=1 intersect

U x Bd(L4) (one in each side) and the intersection of each such member with U x Bd(L4) is exactly two disjoint 2-cells and (2) diam 3 (Ujk) < (l/3)(diam T ). Let t\jk denote the geometric product of fijk 4 with Ai5. Let A, = U! ,;,*=i %», Kl = A Π L and K4 = K{ x ΠΓ=B i*.

6. THEOREM 1. Lβ£ C = ΠΓ=s^ T^e^ C is a Cantor set in

Proof. It follows from the construction that K3, iΓ4, is a decreasing sequence of compact subset of Lr and each K{ is dense in itself. Hence C is dense in itself by Lemma 1. Furthermore, each Ki is a finite union of disjoint compact subsets whose diameters are uniformly small and tend to zero as i —• 00. We conclude then that C is a compact zero-dimensional space which is dense in itself, hence is a Cantor set.

n THEOREM 2. If F is a mapping of AQ x I into L (n ^> 3) such that F I ΛOXO = identity on z/0 and F(AQ x 1) is a point, then F(AQ x I)f]

Proof. The proof is due to [5]. Basically Blankinship had ? n constructed a Cantor set C in AΛ such that C links z/0 in E , hence Λ M An also links AQ in 2£ . As a consequence, JSΓή = iΛ ίl I links Ao in

THEOREM 3. L°° — C has nontrίvial Ist-Homotopy group.

7 Proof. Let F be a mapping of Jo x / into L°° such that i^ 1JQX0 = A WILD CANTOR SET IN THE HILBERT CUBE 193

identity on ΔQ and F(Δ0 x 1) is a point. For each n :> 3, τn(F) is a n mapping of Δo x I into L satisfying (τnF)jQXQ = identity on Jo and

(τnF)(Δ0 x 1) is a point. Hence by Theorem 2, (τnF)(Δ0 x I) Γ) K'n Φ

φ. This implies F(Δ0 x I) Π Kn Φ φ, hence F(J0 x /) Π C ^ ^.

THEOREM 4. There exist two Cantor sets in the Hίlbert cube such that no homeomorphism of one onto the other can be extended to a homeomorphism on the whole Hubert cube.

Let Li = Int(L,) and let (L)~ = IL~=iA. Let VL = K'n n Int(I/)

and Vn = Fή x ΠΓU+i-£; Then each Fw is a closed subset of (L)°°

and hence Co = ΠΓ=3 Vn is both zero-dimensional and closed in (L)°°.

By similar reasoning CQ links Jo in (L)°°. Finally, using the fact s ~

(L)°° and I2~s [2], we conclude:

THEOREM 5. s and l2 contain zero-dimensional closed sets whose complements are not simply-connected.

REFERENCES

1. R. D. Anderson, Topological properties of the Hilbert cube and the Infinite product of open intervals Trans. Amer. Math. Soc. 126 (1967), 200-216. 2. , Hilbert space is homeomorphic to the countable infinite product of lines, Bull. Amer. Math. Soc. 72 (1966), 515-519. 3. , (Abstract), On Extending Homeomorphisms on the Hilbert cube, Notice Amer. Math. Soc, Vol. 13, No. 3, April 1966, p. 375. 4. L. Antoine, Sur Γhomeomorphie de deux figures et de leurs voisinages, J. Math. Pures, Appl. 86 (1921), 221-235.

5. W. A. Blankship, Generalization of a construction of Antoinet Ann. of Math. 53 (1951), 276-297. 6. V. Klee, Convex bodies and periodic homeomorphisms in Hilbert space, Trans. Amer. Math. Soc. 74 (1953), 36.

7. 1 Homogeneity of infinite-dimenssonal parallotopes, Ann. of Math. (2) 66 (1957), 454-460. 8. , Some topological properties of convex sets, Trans. Amer. Math. Soc. 78 (1955), 30-45.

9. f A note on topological properties of normed linear spaces, Proc. Amer. Math. Soc. 7 (1956), 673-674.

Received October 28, 1966, and in revised form May 1, 1967. This paper is a part of the author's doctoral thesis under the direction of Professor R. D. Anderson and revised into its present form while the author held a National Science Foundation Grant GP-5860, at UCLA 1966-1967.

LOUISIANA STATE UNIVERSITY, BATON ROUGE UNIVERSITY OP CALIFORNIA, LOS ANGELES

PACIFIC JOURNAL OF MATHEMATICS

EDITORS H. ROYDEN J. DUGUNDJI Stanford University Department of Mathematics Stanford, California Rice University Houston, Texas 77001

J. P. JANS RICHARD ARENS University of Washington University of California Seattle, Washington 98105 Los Angeles, California 90024

ASSOCIATE EDITORS E. F. BECKENBACH B. H. NEUMANN F. WOLF K. YOSIDA

SUPPORTING INSTITUTIONS UNIVERSITY OF BRITISH COLUMBIA STANFORD UNIVERSITY CALIFORNIA INSTITUTE OF TECHNOLOGY UNIVERSITY OF TOKYO UNIVERSITY OF CALIFORNIA UNIVERSITY OF UTAH MONTANA STATE UNIVERSITY WASHINGTON STATE UNIVERSITY UNIVERSITY OF NEVADA UNIVERSITY OF WASHINGTON NEW MEXICO STATE UNIVERSITY * * * OREGON STATE UNIVERSITY AMERICAN MATHEMATICAL SOCIETY UNIVERSITY OF OREGON CHEVRON RESEARCH CORPORATION OSAKA UNIVERSITY TRW SYSTEMS UNIVERSITY OF SOUTHERN CALIFORNIA NAVAL ORDNANCE TEST STATION

Printed in Japan by International Academic Printing Co., Ltd., Tokyo, Japan aicJunlo ahmtc 98Vl 4 o 1 No. 24, Vol. 1968 Mathematics of Journal Pacific Pacific Journal of Mathematics Vol. 24, No. 1 May, 1968

Harry P. Allen, Lie algebras of type D4 over algebraic number fields ...... 1 Charles Ballantine, Products of positive definite matrices. II...... 7 David W. Boyd, The spectral radius of averaging operators ...... 19 William Howard Caldwell, Hypercyclic rings ...... 29 Francis William Carroll, Some properties of sequences, with an application to noncontinuable power series ...... 45 David Fleming Dawson, Matrix summability over certain classes of sequences ordered with respect to rate of convergence ...... 51 D. W. Dubois, Second note on David Harrison’s theory of preprimes...... 57 Edgar Earle Enochs, A note on quasi-Frobenius rings...... 69 Ronald J. Ensey, Isomorphism invariants for Abelian groups modulo bounded groups ...... 71 Ronald Owen Fulp, Generalized semigroup kernels ...... 93 Bernard Robert Kripke and Richard Bruce Holmes, Interposition and approximation ...... 103 Jack W. Macki and James Sai-Wing Wong, Oscillation of solutions to second-order nonlinear differential equations ...... 111 Lothrop Mittenthal, Operator valued analytic functions and generalizations of spectral theory ...... 119 T. S. Motzkin and J. L. Walsh, A persistent local maximum of the pth power deviation on an interval, p < 1...... 133 Jerome L. Paul, Sequences of homeomorphisms which converge to homeomorphisms ...... 143 Maxwell Alexander Rosenlicht, Liouville’s theorem on functions with elementary integrals ...... 153 Joseph Goeffrey Rosenstein, Initial segments of degrees ...... 163 H. Subramanian, Ideal neighbourhoods in a ring ...... 173 Dalton Tarwater, Galois cohomology of abelian groups ...... 177 James Patrick Williams, Schwarz norms for operators ...... 181 Raymond Y. T. Wong, A wild Cantor set in the Hilbert cube...... 189