THE CANTOR INTERMEDIATE VALUE PROPERTY

Transcription

1 Volume 7, 198 Pages THE CANTOR INTERMEDIATE VALUE PROPERTY by Richard G. Gibso ad Fred Roush Topology Proceedigs Web: Mail: Topology Proceedigs Departmet of Mathematics & Statistics Aubur Uiversity, Alabama 36849, USA topolog@aubur.edu ISSN: COPYRIGHT c by Topology Proceedigs. All rights reserved.

2 TOPOLOGY PROCEEDINGS Volume THE CANTOR INTERMEDIATE VALUE PROPERTY Richard G. Gibso ad Fred Roush J. Stalligs [9] asked the questio: "If oe cosiders I = [0,1] embedded i I x I 1 as I x 0, ca a coectivity fuctio I ~ X be exteded to a coectivity fuctio 1 ~ X?" J. L. Core'tte [3] ad J. H. Roberts [8] gave egative aswers to this questio. A atural questio arises: "I order for the extesio 1 ~ I of a coectivity fuctio I ~ I to be a coectivity fuctio, what is both ecessary ad sufficiet?" Toward this ed we defied the Cator Itermediate Value Property (CIVP) which is give i defiitio. The relatioship betwee the Cator Itermediate Value Property ad this questio is ukow. However we cojecture that if the extesio 1 ~ I of a coectivity fuctio f: I ~ I is a coectivity fuctio, the f has the CIVP. I this paper we give the relatios betwee cotiuous fuctios, coectivity fuctios, Darboux fuctios, ad fuctios havig the CIVP. Defiitio 1. A fuctio f: X ~ Y betwee spaces X ad Y is a coectivity fuctio if ad oly if for each coected subset C of X, the graph of f restricted to C, deoted by flc, is a coected subset of X x Y. The fuctio f is a Darboux fuctio if f(c) is coected for each coected set C.

3 56 Gibso ad Roush Defiitio. A fuctio f: I ~ I has the Cator Itermediate Value Property (CIVP) if ad oly if for ay Cator set K i the iterval (f(x),f(y)) the iterval (x,y) or (y,x) cotais a Cator set C such that f(c) c K. It is well kow that there are Darboux fuctios which are ot coectivity fuctios [1], []. Sice the projectio map is cotiuous, each coectivity fuctio is a Darboux fuctio. Also there are coectivity fuctios which are ot cotiuous. However, every cotiuous fuctio is a coectivity fuctio. To complete the relatios betwee these fuctios we (1) costruct a coectivity fuctio which does ot have the CIVP, () costruct a fuctio which has the CIVP but is ot a Darboux fuctio, ad (3) prove that if f: I ~ I is a cotiuous fuctio, the f has the CIVP. A Hamel basis for the real umbers is a set of umbers a,b,c, such that if x is ay umber the x may be expressed uiquely i the form aa + Sb + yc + where a,s,y, are ratioal umbers of which oly a fiite umber are differet from zero. We ow costruct a Hamel basis for the real umbers which is a subset of I ad has cardiality c i each iterval of positive legth but cotais o Cator set. As stated i [5], Bursti showed the existece of a Hamel basis H which itersects every perfect set of real urr~ers. Let p be i H ad let Hip be the elemets of H divided.by p. The 1 is the oly ratioal umber i Hip ad Hip is a Hamel basis for the reals.

4 TOPOLOGY PROCEEDINGS Volume Lemma 1. Hip itersects every perfect subset of the reazs. Proof. Suppose P is a perf~ct set. The pp is a perfect set ad pp H ~~. Let y be i H ad a be i P such that pa = y. The a = yip is i Hip. So a is i P Hip. Let G = {y - [y]: y is i Hlp ad y ~ I} U {I} where [y] is the greatest iteger less tha or equal to y. The G is a subset of I. Sice Hip I is a subset of G ad itersects every perfect subset of I, G does also. Also G is a Hamel basis for the reals. Lemma. G cotais o perfect subset. Proof. Suppose P is a perfect subset of G. The choose a ratioal UITIDer q such that q lies betwee two poits of P. So q ~ 1. Let M = P [O,q] ad let N = P 1 [q,l]. Now M ad N are perfect sets. Assume q ~~. The M + (l-q) is a perfect subset of I. So there exists a x i G such that x is i M + (l-q). Let x = y + (l-q) where y is i M. Sice x - y + (q-l) 0, it follows that l,x,y are liearly depedet which is a cotradictio. Similarly we have a cotradictio, if q > ~. Lemma 3. Every Cator set cotais c disjoit sub- Cator sets. Proof. Let K be a Cator set. The for each x i K, {x} x K is a Cator set. If x ~ y, the ({x} x K) ({y} x K) =~. Thus K cotais c disjoit sub-cator sets. Sice K is a Cator set ad ay two Cator sets are homeomorphic, K cotais c disjoit sub-cator sets.

5 58 Gibso ad Roush Sice a Cator set cotais c disjoit sub-cator sets, if a set meets every perfect subset i a iterval it has cardiality c there. Thus G has the desired properties. The First Example. Let f = {(x,o): x is ot i G}. l Let K be the collectio of closed subsets of I x I such that ITX(K) has cardiality c for each K i K where TI x x-projectio. f of 1 x I such that is the Usig trasfiite iductio, select a subset (1) f itersects each elemet of K ad () if x ad yare i f, the TIx(x), TIx(y) are i G ad ITx(x) ~ TIx(y). Let f f l U f U {(t,l): t is i I - TIx(f l U f )}. The I ad f is the graph of a fuctio f: I + I. Suppose f is ot a coectivity fuctio. The f is the sum of two mutually separated sets A ad B ad there exists two mutually exclusive ope sets U ad V i 1 x 1 such that A c U ad B c V. Let K = I x I - U U V. The K cotais a cotiuum L that separates I x I. Choose poits P ad Q i L such that TIx(P) ~ TIx(Q). The there exists (z,o) i f l ad E > such that (1) the disc {(x l,x ) E 1 : IX I - zl < ad x < } K = ~ ad () the iterval {xl E 1 : IX I - zl < } is betwee TIx(P) ad TIx(Q). Let S = {(x l,x ) E 1 : z +. xl < z + ad 0 < x. I}. The K S c 1 is closed ad (K S) f = ~. So TIx(K S) has cardiality less tha c. Thus there exists a u such that z + < U < z + ad u is ot i TIx(K S). Hece

6 TOPOLOGY PROCEEDINGS Volume { (u,t): 0 ~ t I} separates P from Q i 1 ad does ot itersect L. This is a cotradictio. Hece ITx(P) = ITx(Q) ad L is a proper subset of a vertical iterval of I. So L does ot separate r. This is a cotradictio. Therefore r is coected ad f is a coectivity fuctio. We ow show that f does ot 'have the CIVP. Choose f(x) ad fey) i I such that f(x) ~ fey). A~x < yad f (x) < f (y). Let C be a Cator set~/ ope iterval (f(x),f(y». Now G (x,y) c6i{tais o Cator set but is of cardiality c. So there exists o Cator set i (x,y) which maps ito C. Defiit~o 3. A subset ScI is Cator dese if ad oly if for ay 0 < a < b < 1, [a,b] S cotais a Cator set. Lemma 4. I is the czosure of the uio of Catop dese subsets St for each t i I ad Sr Ss _ if r ~ s. Proof. Let KI,1 be a Cator set i [o,~] ad let K I, 1 be a Cator set i [,1] such that Kl,l K l, Let K,~ be a Cator set i [O,}], let K, be a Cator. [1 1] 1 b C. [1 3] d 1 set l ~,~, et K,3 e a ator set l ~,~, a et K,4 be a Cator set i [i,l] such that the collectio K 1 =1, or m == 1,,3,4 are pairwise disjoit. I,m Cotiuig this process, let K 1 be a Cator set i, [O,1/ ], K, be a Cator set i [1/,/ ],.~., let K, be a Cator set i [(-l)/~,1] such that Kl,m Kp,q = ~, if 1 ~ p or m ~ q. _. ad Now decompose each Kl,m ito c disjoit Cator sets

7 60 Gibso ad Roush t sets K for each t i I. Let S = U K t The St is l,m t I,m I,m Cator dese i I. Sice St is dese i I, the closure of UtS is equal to I. t The Secod ExampZe. Let g: I + {irratioal umbers i I} be 1-1 ad oto. Defie f(x) g(y) where x is i S. Y If x is ot i S for each y i I, let f(x) = o. y Let a ad b be i I such that a < b. Suppose f(a) < f(b). Let K be a Cator set i (f(a),f(b)) ad let z be a irratioal umber i K. Let w = g-l(z). Cosider S. If w x is i Sw' the f(x) = g(w) z. Thus Sw maps ito K. By Cator desity there is a Cator set C c Sw [a,b]. Thus!(C) c K ad f has the CIVP. Clearly f is ot a Darboux fuctio. Theopem 1. If f: I + I is cotiuous~ the f has the CIVP. Ppoof. Assume x ad yare i I ad x < y. Suppose f(x) < f (y) Let C be a Cator set i (f(x),f(y»). Let K = f-1(c) (x, y). Sice f(x) ad f (y) are ot i C, x ad yare ot i f -1 (C). Hece K f-l(c) [x, y] ad K is closed. Sice K is closed ad has cardiality c, K cotais a Cator set P. Sice f(p) c C we are doe. Theorem. If f: I + I is a czosed fuctio which has the CIVP~ the f is a cotiuous fuctio. Proof. Choose a ad b i I such that a < b. Assume f(a) < f(b). Suppose f is ot a Darboux fuctio. The there exists a y i the ope iterval (f(a),f(b)) such that if x is i the iterval (a,b), the f(x) ~ y.

8 TOPOLOGY PROCEEDINGS Volume Choose a positive iteger N such that [y - ~,y + ~] is a subset of (f(a),f(b)). Now for each Cator set C c (f(a),y - ~), there exists a Cator set K c {a,b) such that f(k) c C. Choose p i K. The f(p) is i 1 (f(a),y - N). Likewise we have a q i (a,b) such that f(q} 1 is i (y + N,f(b)). Assume p < q. that: We ca costruct a collectio C of Cator sets such 1,,3,- - -, (1) C c [y N+ 1} N+:'y c (f(p),f(q)) where ~ () Y is i each C ' (3) x is i [p,q}, f(x ) is i C ' ad x ~ x if m ro, ad (4) f(x i ) is ot i C if i = 1,,---,-1. Now C = {y} ad f(x ) coverges to y. There exists a subsequece {a } of {x } such that a coverges to some x i [p,q} c (a,b). Now {a } U {x} is a closed set. Hece f({a } U {x}) is closed. Sice f(a ) coverges to y ad y is ot i the set f({a } U {x}) we have a cotradictio. Therefore f is a Darboux fuctio. Sice a closeq Darboux fuctio is cotiuous, f a cotiuous fuctio. is We make the followig remarks. (1) As stated by the referee the fuctio i the first example is almost cotiuous [7}. A fuctio f i~ said to be almost cotiuous if each ope set cotaiig f also cotais a cotiuous fuctio with the same domai. Stalligs [9} showed that if the fuctio f is almost

9 6 Gibso ad Roush cotiuous ad has coected domai, the f is a coectivity fuctio. Examples of coectivity fuctios which are ot almost cotiuous are give i [1], [3], [6], ad [8]. Clearly every cotiuous fuctio is almost cotiuous. Thus with previous results ad this remark the relatios betwee cotiuous, almost cotiuous, coectivity, Darboux, ad fuctios havig the CIVP are kow. () It follows from lemma 3 that there exists Cator sets which cotai o ratioal umbers. Refereces 1. J. B. Brow, Coectivity~ semi-cotiuity~ ad the Darboux property, Duke Math. J. 36 (1969), A. M. Brucker ad J. G. Ceder, Darboux cotiuity, Jber. Deutsch. Math.-Verei. 67 (1965), J. L. Corette, Coectivity fuctios ad images o Peao cotiua, Fud. Math. 58 (1966), o. H. Hamilto, Fixed poits for certai o-cotiuous trasformatios, Proc. A.M.S. 8 (1957), F. B. Joes, Measure ad other properties of a Hamel basis, Bull. A.M.S. 48 (194), ad E. S. Thomas, Jr., Coected Go-graphs, Duke Math. J. 33 (1966), K. R. Kellum ad B. D. Garrett, Almost cotiuous real fuctios, Proc. A.M.S. 33 (197), J. H. Roberts, Zero-dimesioal sets blockig coectivity fuctios, Fud. Math. 57 (1965), J. Stalligs, Fixed poit theorem for coectivity maps, Fud. Math. 47 (1959), Columbus College Columbus, Georgia ad Alabama State Uiversity Motgomery, Alabama 36101