A FINITE TO ONE OPEN MAPPING PRESERVES SPAN ZERO

Transcription

1 Volume 13, 1988 Pages A FINITE TO ONE OPEN MAPPING PRESERVES SPAN ZERO by Mara Cuervo and Edwn Duda Topology Proceedngs Web: Mal: Topology Proceedngs Department of Mathematcs & Statstcs Auburn Unversty, Alabama 36849, USA E-mal: topolog@auburn.edu ISSN: COPYRIGHT c by Topology Proceedngs. All rghts reserved.

2 TOPOLOGY PROCEEDINGS Volume A FINITE TO ONE OPEN MAPPING PRESERVES SPAN ZERO Mara Cuervo and Edwn Duda It s shown that span zero s preserved under fnte to one open mappngs. 54CIO, 54F20 Introducton The noton of span of a compact metrc space and ts natural generalzaton semspan were defned by A. Lelek n 1964 and 1977, respectvely [6, 7J. It follows from the defntons that semspan s greater than or equal to span and both functons are monototne wth respect to closed subsets. It can be shown drectly that a nonuncoherent contnuum or a trod have span greater than zero. All chanable contnua have semspan zero and those contnua wthout the fed pont property have span greater than zero. J. F. Davs has shown that span zero and semspan zero are equvalent [2J. In [llj I. Rosenholtz has shown that an open mage of a chanable contnuum s also chanable. A natural queston s whether open mappngs preserve span zero. Notatons and Defntons A metrc contnuum s a compact connected metrc space. We let d represent the dstance functon and IT I and IT 2 the natural projecton mappngs of the Cartesan Product X X onto X. The semspan of a compact metrc

3 182 Cuervo and Duda space, denoted by 00(X), s the least upper bound of all real numbers E such that there s a subcontnuum Z of X X wth the propertes that IT (Z) C ITl(Z) and d(,y) > E for 2 all (,y)ez. The defnton of span of X, denoted by o(x), S,the same ecept that we requre IT 2 (Z) = ITl(Z). A mappng f(x) = Y s weakly confluent provded that for each subcontnuum Key, there s a component H of f-l(k) such that f(h) = K. A contnuum Y s n class W provded that for each mappng f from a contnuum onto Y, f s weakly confluent. A contnuum s ndecomposable provded t cannot be the unon of two proper subcontnua. For EX, defne K as follows; K = n{k, K s a subcontnuum of X and s n the nteror of K}. The propertes of for certan contnua are developed n [IJ. A sub- K contnuum B s termnal n the space X f whenever Hand K are two subcontnua wth B n H ~ ~ and B n K ~ ~, then B U H C B U K or B U H ~ B U K. Prelmnary Results A mappng s fnte to one f each pont nverse s a fnte set. Lemma 1. Let f(x) = Y be a fnte to one open mappng~ where X and Yare looally oompaot separable metro spaoes. The set D of all ponts of X suoh that f s a looal homeomorphsm at eaoh pont of f-lf() s an open dense subset of.

4 TOPOLOGY PROCEEDINGS Volume Proof Let D = { E X and f-lf() has less than n or equal to n ponts}. Each D s a closed set by the n openness of f. Snce X = n~ldn and X s a Bare Space some D contans nteror ponts. Let be the least n nteger for whch D has nteror ponts and let be an nteror pont. Each pont of f-1f() s nteror to D snce f-lf(nt.m ) s an open set. Let f -1 f() = {l' 2 ', } and take parwse dsjont open sets U( ), = 1,2,..,n about each pont of f- 1 f() and contaned O n nt.m Now f I U(X ), = 1,2,..,n s an open and O 1-1 mappng of U( ) onto f(u( )) and s therefore a homeomorphsm. To show D s dense n X apply the argument above to an arbtrary non-empty open set U. That s 00 U n~l(u n D ) and U as a subspace s a Bare space so n there s a least nteger such that U n D has nteror ponts. See [12J, VII, 3.5 for a smlar result. Lemma 2. If f(x) = Y s a fnte to one open mappng, where X and Yare locally compact separable metrc spaces, and B s a closed subset of X such that nt.f(b) s not empty, then nt.b s not empty. Proof. By the precedng result there s a pont Yo E nt.f{b) such that f s a local homeomorphsm at

5 184 Cuervo and Duda -1 each pont of f (YO). and choose parwse dspont open sets U(), so that W = f(u(.) = f(u(.» 1. J C nt. feb) for all 1,,,j = 1,.,n and each flu( ) s a homeomorphsm of O U( ) onto f(u(». We have W ~lf(b n U(» and each feb n U(X» s an Fa set and W as a subspace s a Bare space so some feb n U(X has nteror ponts. If V s» an open set n feb n U(X then the set U n B n U(X )» whch maps onto V s open n X. Lemma 3. Let X be a heredtarly unaoherent atroda metra contnuum. For E X, f K has non-empty nteror then K s ndecomposable or the unon of two ndecom posable aontnua wth non-empty nterors. Proof. Suppose K = L U M, where Land M are proper subcontnua. We can assume L r-=:m and M = M - L. By the defnton of K t follows that ~ L - M and ~ M - L so E L n M and furthermore E nt.(l U M) otherwse the space X contans a trode Suppose L = LIU L 2, where L l and L are proper subcontnua. If 2 ~ L then E nt.(l U M) whch contradcts the defnl, 2 ton of K so that E L n L n M ths mples L U L U l 2 l 2 M s a trod and ths s not possble hence Land Mare ndecomposable wth non-empty nterors. From [3] we have: Theorem 2. If 00(X) = E > 0, there ests an ndecomposable contnuum I C X wth

6 TOPOLOGY PROCEEDINGS Volume (1) = E and every proper subcontnuum of I has semspan less than E. The space X n ths result must be an atrodc heredtarly uncoherent metrc contnuum. Man Result artcle. The followng theorem s the man result of ths Theorem. Let X and Y represent metrc contnua. If 00(X) = 0 and f(x) = Y s a fnte to one open mappng, then 00(Y) o. Proof 0o(X) o mples X s atrodc and heredtarly uncoherent. By [8J 00(X) = 0 mples X s treelke and by [5J atrodc and tree-lke mples X s n class W. By [9J Y s heredtarly uncoherent and by [10J Y s tree-lke. The mappng f takes atrodc contnua onto atrodc contnua, so by [5J, Y s n class W. By Theorem 2, f 00(Y) = E > 0, then there ests an ndecomposable subcontnuurn I wth 00(1) = E and every proper subcontnuurn of I has semspan less than E. Let H be a -1. component of f (I), then, as s well known, f(h) = I and f/h s a fnte to one open mappng of H onto I. For E H, K = nk a, K C H, a E r and E nt K a relatve to a H. The contnuum f(k a ) C I has nteror ponts so f(r O ) = I snce I s ndecomposable. Thus, n ths case, we K can argue that f(k ) = I and snce f s fnte to one and open, by Lemma 2 nt K 1=~. There are only two possble cases; () K s ndecomposable, or () K = K U K ' where y z K and K have non-empty nteror, are ndecomposable, y z

7 186 Cuervo and Duda nt K n nt K ~ and E K n K. Ths means that, y z y z snce f s fnte to one and open on H, we can consder H as a fnte lnear chan of ndecomposable contnua,.e., H = K U K U,., UK, where {nt K }, l 2 n = l,..,n s a parwse dsjont collecton and only successve K. 's ntersect. The contnuum H s rredu ~ cble from a pont p E K to a pont q.e K There s l n a contnuum Z C I I such that for all (,y) E Z, d(,y) > E and ITl(Z) = I = IT (Z). The contnuum Z can be 2 chosen so that t s ndecomposable and ITI/Z and TI /Z 2 are rreducble mappngs [4J. Let C be the composant of K whch s accessble from K and let L be the com X 1 2 posant of I whch contans fcc). The composant L may be epressed as, L D s a contnuum. For each, Z - TI~l(D} s open n Z -1 and connected and dense snce IT I (D ) cannot meet all composants of Z. by a Bare theorem and hence there ests an (,y) E Z wth,y E I - L. Snce (f f) (K K ) = I I there l l s a pont (a,b) E K K wth (f f) (a,b) (,y) l l and a,b E K - C. Let B be the component of (f f)-l(z) l whch contans (a,b). We have f(itl(b»

8 TOPOLOGY PROCEEDINGS Volume nteror ponts and thus cannot be a proper subcontnuum Smlarly IT (B) ~ K. Snce K s a termnal subcon 2 l l tnuum n H t follows that TI (B) ~ TI (B) or TIl(B) C IT (B) Snce B does not meet the dagonal of H, we have 00(H) > 0 and consequently 0Q(X) > 0, whch s a contradcton. References 1. Cuervo, M. and Duda, E., A characterzaton of Span Zero, Houston Journal of Math. v. 12 (1986). 2. Davs, J. F., The equvalence of zero span and zero semspan, Proc. Amer. Math. Soc. V. 90 (1984), Duda, E., A characterzaton of semspan of contnua, Proc. Amer. Math. Soc. V. 96 (1986), , Contnuum Coverng Mappngs, Proc. of Sth Prague Topol. Symp. (1986). 5. Grspo1aks, J. and Tyanchatn, E. D., Contnua whch are mages of weakly co~fluent mappngs only, Houston Journal of Math., V. 5 (1979), Lelek, A., Dsjont Mappngs and Space of Spaces, Fund. Math. V. 55 (1964), , On the Surjectve Span and Semspan of Connected Metrc Spaces, Colloquum Math. V. 37 (1977), ' The span of mappngs and spaces, Topology Proc. V. 4 (1979), , On Confluent Mappngs, Colloquum Math. V. 15 (1966), McLean, T. B., Confluent Images of tree-lke curves are tree-lke, Duke Math. J. V. 39 (1972),

9 188 Cuervo and Duda 11. Rosenholtz, I., Open maps of ahanabze aontnua, Proc. Amer. Math. Soc. V. 42 (1974), Whyburn, G. T., TopoZogaaZ AnaZyss, Prnceton Unversty Press, gnversty of Mam Coral Gables, Florda 33124