The Property of Kelley and Confluent Mappings

Transcription

1 BULLETIN OF nte POLISH ACADEMY OF SCIENCES Mathematics Y0l. 31 No , 1983 GENERAL TOPOLOGY The Property of Kelley and Confluent Mappings by Janusz J. CHARA TONIK Presented by K. URBANIK on April 9, 1983 Summary. In 1977 R. W. Wardle introduced and investigated a pointwise version of the property of Kelley. ln this paper a preservation is studied of the property under mappings which satisfy some local conditions (e.g. confluency relative to a point). As a consequence of obtained results we prove that continua which are homogeneous with respect to open mappings have the property of Kelley. A continuum means a compact connected metric space. A continuum X (with a metric d) has the property of Kelley provided that given any e > 0 there exists a b > 0 such that for each two points a and b of X satisfying d (a, b) < (j and for each subcontinuum A of X containing the point a there exists a subcontinuum B of X containing the point b and satisfying dist (A, B)< e (here dist denotes the Hausdorff distance in the hyperspace C (X) of all non-empty subcontinua of X). In [2], (16.38), p. 559 Sam B. Nadler, Jr. asks "What kinds of mappings preserve the property of Kelley?" In the present paper we discuss a similar question for the property of Kelley considered as a local property (see [3]). Answers are formulated in terms of confluency of the mapping relative to a point of the domain space. The majority of the results obtained in the paper form pointwise versions of some global properties proved in [3]. Some,of them are applied at the end of the paper to homogeneous continua. Namely Theorem 2.5 of [3], p. 293 (which states that homogeneous continua have the property of Kelley) is generalized to continua that are homogeneous with respect to open mappings. Thanks are due to W. J. Charatonik for helpful discussions on the topic of this paper. Iff: X--+ Y is a mapping between continua X and Y, then the mapping fc (X)-+ C (Y) induced by f is defined by /(A)={! (12):aEA}. It can be observed that continuity off implies one off The mapping/: C 2 (X)--+ C 2 (Y) (where C 2 (X)= C (C (X))} induced by f will also be considered.

2 376 J. J. Charatonik ForeachaeX we let rxx(a)= {AEC(X):aEA}. It is shown in [3], p. 292 that rxx:x--+ C 2 (X), and that rxx is upper semi-continuous at each point of X. Recall that a continuum X is said to have the property of Kelley at a point a EX (see [3], p. 292) provided for each e > 0 there is a b > 0 such that if be X with d (a, b)< b and if aeaec (X), then there exists a B with bebec (X) and dist (A, B)< e. Clearly a continuum has the property of Kelley if and only if it has this property at each of its points. The following proposition is known (see [3], Theorem 2.2, p. 292). PROPOSITION A. The function rxx is continuous at a point aex if and only if X has the property of Kelley at a. Let two continua X and Y be given. A mapping f: X --+ Y is called confluent provided every component of the inverse image of a continuum Q c Y is mapped onto Q under f. A mapping f: X--+ Y is called confluent relative to a point a EX if for each continuum Q c Y such that f (a) E Q the component of f- 1 (Q) containing a is mapped onto Q under f. Note that (1) A mapping is confluent if and only if it is confluent relative to each point of its domain. Now consider the following diagram. c 2 (X) L C 2 (Y) ax j iay (2) X / y The next proposition, which concerns the above diagram, i!> a pointwise version of Theorem 4.2 of [3], p. 296, and its proof is similar to. one of the global case. We include it for the sake of completeness only. PROPOSITION 1. Diagram (2) commutes at a point a EX, i.e., the equality (3) /(rxx (a))= rxy (f (a)) holds, if and only if the mapping f is corifluent relative to a. Proof. By the definitions of/, rxx and rxy, equality (3) is equivalent to (4) {f (A):aEAEC (X)}= {BEC (Y):f (a)eb}. Continuity of f implies that the left member of (4) is a subset of the right one. To prove the inverse inclusion take a BEC(Y) with f(a)eb. Then the needed continuum A in X is the component of f- 1 (B) containing the point a. Now assume the diagram commutes at a, i.e., (4) holds. Take BEC (Y) with f (a)eb. By (4) there exists an A E C (X) with aea such that B = f (A). Let C be the component of f- 1 (B) containing a. Since A is connected and contains a, we have A c C, whence B =f(a) cf(c) c B, and consequently f (C)= B. Proposition 1 and (1) imply CoROLLARY 1 ([3], Theorem 4.2, p. 296). Diagram (2) commutes if and only iff is confluent.

3 The Property of Kelley and Confluent Mappings 377 It is proved in [3], Theorem 4.3, p. 296 that if a continuum X has the property of Kelley and Y is the image of X under a confluent mapping, then Y has the property of Kelley, too. We are interested in the following problem: under what conditions on a mapping f:x--+ Y the property of Kelley at a point a EX is preserved at a point f (a) of the range space Y? A simple pointwise version of that theorem quoted above is not true. Namely we have ExAMPLE 1. There exist a smooth fan X having the property of Kelley at a point ae X and a confluent mapping f of X onto a smooth fan Y which does not have the property of Kelley at f (a). In fact, take in the cartesian plane R 2 two countable sets A= {(1, 0)} u u{(1, 1/n):nEN} and B= {(-1,0)} u{(-1/2, 1/n):nEN}, where N is the set of natural numbers, and let X be the cone with the vertex (0, 0) over Au B. Thus X is a smooth fan having the property of Kelley at a= (1/2, 0). Define a mapping f:x--+ R 2 of X into the plane putting f((x,y))=(x,y) if x o:o andf((x,y))=(-x, -y) if x<o. Thus Y=f(X) is again a smooth fan, the mapping f of X onto Y is confluent, but Y does not have the property of Kelley at f (a)= a. Note that, in the above example, the continuum X does not have the property of Kelley at ( -1/2, 0), the point which belongs to f - 1 (f (a)). This fact is essential, because if X has the property of Kelley at each point of f- 1 (f (a)), where f: X--+ Y is confluent, then Y has this property at f (a). Even a stronger result is true. To formulate it we need one more definition. A mapping f: X --+ Y is said to be confluent at a point y E Y provided for every continuum Q containing y, each component o(f-- 1 -(Q) fsmapped onto Q under f. Obviously, (5) a mapping f:x--+ Y is confluent at a point ye Y if and only if it is confluent relative to each point of the set f- 1 (y). The following proposition holds which is due to W. J. Charatonik. PROPOSITION 2 (W. J. Charatonik). If a mapping f:x-+ Y of a continuum X onto Y is confluent at a point Yo E Y and if X has the property of Kelley at each point off - 1 (y 0 ), then Y has the property of Kelley at y 0. Proof. We have to show that (6) for each e > 0 there is a b > 0 such that if ye Y with d (y, y 0 )'< b and if y 0 EPeC (Y), then there exists a Q with ye_qec (Y) and dist (P, Q) <e. So, let e > 0 be given. Since f: C (X)--+ C ( Y) is continuous, for this e there is an e' > 0 such that for each two subcontinua K and L of X, if dist (K, L) < e', then dist (/(K), /(L)) <e. Further, since X has the property of Kelley at each point a of f- 1 (y 0 ), for this e' there is a (/ > 0 such that for every x EX and for every K with a EKE C (X) there exists an L with xelec (X) such that d (x, a)< b' implies dist (K, L) < e'. The

4 378 J. J. Charatonik set f -l (y 0 ) being compact, the number b' can be choosen uniformly for all points aef- 1 (y 0 ). Let b" be this number. Since f- 1 : Y-+ 2x is upper semi-continuous (see [1], 43, I, Theorem 1, p. 57 and II, Theorem 1, p. 61), for this b" there exists a b > 0 such that for every y E Y if d (y, y 0 ) < [J, thenf~ 1 (y) c N {f- 1 (y 0 ), b"), where N (S, 17) stands for the 17-neighborhood about a set S. We show that b satisfies (6). Indeed, take an arbitrary point ye Ywith d (y, y 0 ) < [J, and let P be such that y 0 EPEC (Y). By upper semi-continuity of f- 1 there are points xef- 1 (y) and x 0 Ef- 1 (y 0 ) such that d(x,x 0 )<b". Let K be the component off- 1 (P) with x 0 EK. Then by confluency off at Yo we have j(k) = P. Since X has the property of Kelley at x 0, there exists an L with xelec (X) and dist (K, L) < e'. Then by continuity off we have dist (j(k), j(l)) <e. Put Q = /(L). Then y E Q E C ( Y) and dist (P, Q) < e. The proof is complete. CoROLLARY 2 ([3], Theorem 4.3, p. 296). If a continuum X has the property of Kelley, and a continuum Y is a confluent image of X, then Y has the property. of Kelley. Observe that, according to (5), confluency off relative to each point of f- 1 (y 0 ) is assumed in Proposition 2. This assumption is essential. Namely take B'={(-1/2,0)}u{(-1/2,1/n):nEN} and let X' denote the cone with the vertex (0, 0) over Au B', where A is as in Example 1. Thus X' is a smooth fan having the property of Kelley at each point. Defining the mapping f exactly in the same way as in Example 1 we see that f is confluent relative to (1/2, 0) but is not confluent relative to (-1/2, 0), and both these points are mapped onto (1/2, 0), the point at which Y does not have the property of Kelley. Proposition 2 is not a satisfactocy answer to the question on preservation of the property of Kelley at a point under a mapping, because its hypotheses concern not only the point a, but all points of f- 1 (f (a)), as regards both the property of Kelley and the confluency of the mapping relative to appropriate points. From this point of view the next proposition is better. To formulate it we recall one notion and two equivalences. A mapping f: X -+ Y is said to be interior at a point a E X provided for every open neighborhood U of a in X the point f (a) is in the interior off (U). Obviously we see that (7) a mapping is open if and only if it is interior at each point of its domain. Note that the property of Kelley at a point can be formulated in the following way. (8) A continuum X has the property of Kelley at a point a EX if and only if for every subcontinuum A c X containing a and for every sequence of points an EX converging to a there exists a sequence of continua An c X converging to A such that anean. PROPOSITION 3. If a continuum X has the property of Kelley at a point

5 The Property of Kelley and Corifluent Mappings 379 a EX and if a surjective mapping f: X--+ Y is both interior at a and confluent relative to a, then Y has the property of Kelley at the point f (a). Proof. Let f (a)ebec (Y) and let a sequence of points b,e Y converging to f (a) be given. If A denotes the component of f- 1 (B) containing a, then f (A)= B by confluency off relative to a. Take an arbitrary open neighborhood U about a in X. Since f is interior at a, the point f (a) is in the interior of f (U), so almost all points b, are in f (U). Thus there is a sequence of points a,e U such that f (a,)= b,. Taking a suitable subsequence if necessary we may assume a, converges to a. Since X has the property of Kelley at a, there is a sequence of subcontinua A, of X converging to A and such that a,ea, for every n. Letting B, = f (A,) we get a sequence of continua B, containing b, and converging to B. This completes the' proof. Now Propositions A, 1 and 3 imply CoROLLARY 3. If a continuous mapping f of a continuum X onto Y is both interior at a point a EX and confluent relative to a, and if the JUnction rx x: X --+ C 2 (X) is continuous at a, then diagram (2) commutes at a and the JUnction rx.y: Y--+ C 2 (Y) is continuous at f (a). Observe that an open mapping f: X --+ Y of a continuum X onto Y is both interior at a point a EX (see (5)) and confluent ([ 4], (7.5), p. 148), thus confluent relative to a. Therefore Proposition 3 leads to CoROLLARY 4. If a continuum X has the property of Kelley at a point a EX and if a mapping f of X onto Y is open, then Y has the property of Kelley at f (a). Note that openness of f in Corollary 4 cannot be relaxed to its confluency, as Example 1 shows. Further, the two assumptions on f in Proposition 3 are both essential. In fact, on the one hand, the mapping f: X--+ Y defined in Example 1 is confluent, thus confluent at a = (1/2, 0), but not interior at this point, and the property of Kelley of X at a is not transferred to Yatf (a)= a under f. On the other hand, the mapping f of the well known sin (1/x)-continuum onto sin (1/x)-circle, defined as the identification of the end point (1, sin 1) with an end point of the limit segment, is interior at the middle point a= (0, 0) of the limit segment, but is not confluent at this point, and again the property of Kelley is not transferred. One can ask if the property of Kelley of a continuum at a point is preserved in the opposite direction, i.e., if theorems inverse to Proposition 3 or to Corollary 4 are true. The answer is negative. Namely we have ExAMPLE 2. There is a dendroid X and an open mapping f of X onto a smooth fan Y such that Y has the property of Kelley at a point y 0, while X does not have this property at the only point of f- 1 (y 0 ). Really, take in the cartesian plane a harmonic fan Y, and let y 0 be an end point of the. limit segment opposite to the vertex of the fan.

6 380 J. J. Charatonik Let Y* be the image of Y under the central symmetry with respect to y 0, and put X = Y u Y*. Define f: X-+ Y by f I Y-the identity and f I Y*-the central symmetry. thus f is open, Y has the property of Kelley at each point, in particular at y 0, while X does not have the property at this point. Recall that a space X is said to be homogeneous with respect to a class M of mappings if for each two points a and b of X there exists a continuous surjection f:x-+ X such that f EM and f (a)= b. In particular, if M is the class of homeomorphisms, we get the well known concept of a homogeneous space. Theorem 2.5 of [3], p. 293 states that homogeneous continua have the property of Kelley. We prove a more general result from which the theorem of Wardle follows as a corollary. Namely we have the following STATEMENT. If a continuum is homogeneous with respect to the class of open mappings, then it has the property of Kelley. In fact, Theorem 2.3 of [3], p. 293 says that any continuum has the property of Kelley at each point of a dense G.,-set. Thus the statement follows from Corollary 4. Taking into account Wardle's result (see Corollary 2) the following question seems to be natural. QuEsTION. Does homogeneity with respect to the class of confluent mappings imply the property of Kelley for continua? INSTITUTE OF MATHEMATICS UNIVERSITY PL. GRUNWALDZKI 2/4, WROCLAW (INSTYTUT MATEMATYKI, UNIWERSYTET WROCLAWSKI) REFERENCES [1] K. Kuratowski, Topology, vol. 2, Academic Press and PWN, [2] S. B. Nadler, Jr., Hyperspaces of sets, Marcel Dekker, Inc., [3]. R. W. Wardle, On a property of J. L. Kelley, Houston J. Math., 3 (1977), [4] G. T. Whyburn, Analytic Topology, A.M. S. Colloq. Publications, Jl. E. XapaTOHHK, CBOicTBO KeJIJIII R KoutJIIo:BITuye otofipaxcehrii B 1977 r. Yop;:w sseji u uccjie.uosaji TO'Ie'IHYIO sepcu10 csoii:ctsa KeJIJIH. B HaCTOlUIJeii: pa6ore Mbi HCCJie.UyeM COXpaHeHHe :HOfO CBOHCTBa OTHOCHTeJIHO OT06palKeHHH HCIIOJIHlllOIIJHX HeKO TOpbie JIOKaJibHb!e YCJIOBI!ll (HarrpHMep KOH<jJJIIOJHTHOCTb OTHOCI!TeJihHO TO'IKH) 8 Ka'leCTBe CJie.!(CTBHll H3 IIOJIY'IeHbiX pe3yjibtatob.!(oka3yem, 'ITO KOHTHHYYMbi KOTOpble O.!(HOpO.UHbl OTHO CHTeJihHO OTKpbiTbiX OT06paJKeHHH 06Jia.!(alOT CBOHCTBOM KeJIJIH.