SOME REMARKS ON FREELY DECOMPOSABLE MAPPINGS

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1 Volume, 1977 Pages SOME REMARKS ON FREELY DECOMPOSABLE MAPPINGS by C. Bruce Hughes Topology Proceedigs Web: Mail: Topology Proceedigs Departmet of Mathematics & Statistics Aubur Uiversity, Alabama 36849, USA ISSN: COPYRIGHT c by Topology Proceedigs. All rights reserved.

2 TOPOLOGY PROCEEDINGS Volume SOME REMARKS ON FREELY DECOMPOSABLE MAPPINGS c. Bruce Hughes 1. Itroductio Freely decomposable mappigs were recetly itroduced by G. R. Gordh, Jr. ad the author i [1] as a geeralizatio of mootoe mappigs. It was show that every iverse sequece of locally coected (semi-locally coected) cotiua with freely decomposable bodig mappigs has a locally coected (semi-locally coected) limit. Other basic properties of freely decomposable mappigs were established i [1]. For example, every freely decomposable mappig of a uicoheret cotiuum oto a locally coected cotiuum is mootoe. This paper cotiues the study of freely decomposable mappigs. A cotiuum is a compact coected metric space ad a mappig is a cotiuous surjectio betwee cotiua. If X is a cotiuum, the X = A U B is a decompositio provided that A ad B are proper subcotiua of X. A mappig f: X + Y is said to be freely decomposable (deoted FD) if for each decompositio Y = A U B there exists a decompositio X = AI U B ' such that f(a ) ~ A ' ad f (B I) ~ B. The cotiuum X is free Zy decomposable if for each pair of distict poits a ad b i X, there exists a decompositio X = A U B such that a E A'B ad b E B\A. It is kow that a cotiuum is semi-locally coected if ad oly if it is

3 14 Hughes freely decomposable [].. A Certai Class of Cotiua I [1] it was show that every FD mappig oto a locally coected cotiuum which cotais o separatig poits is mootoe. The followig problem aturally arises. ProbZem. Charac~erize those cotiua oto which every FD mappig is mootoe. This class of cotiua the icludes all locally coected cotiua without separatig poits. The figure eight is a example of a cotiuum i this class which cotais a separatig poit. As the ext two theorems show ay cotiuum Y i this class is locally coected ad cotais o arc each poit of which separates Y. Theorem 1. If Y is a o-zocazzy coected cotiuum 3 the there exist a cotiuum X ad a o-mootoe FD mappig f: X -+ Y. Proof. Sice Y is ot locally coected, there exist a poit p E Y, a ope set U ~ Y cotaiig p, a cotiuum K such that p E K ~ ct(u) ad K bd(u) ~ ~, ad a sequece {C } of distict compoets of U disjoit from K such that K = lim{c }. Let K' be a topological copy of K ad let h: K -+ K' be a homeomorphism. Let X = Y U K' be the cotiuum obtaied by.. attachig Y to K' alog K bd(u) by the restrictio of h to K bd(u). Now defie f: X -+ Y by E Y, ad f(x) =ix if x h-l(x) if x E K'. It is clear that f is a mappig, ad f is ot mootoe sice

4 TOPOLOGY PROCEEDINGS Volume f (p) = {p,h(p)}. To see that f is a FD mappig, let Y = A U B be a decompositio. Let A' be the compoet of f-l(a) which cotais f-l(a) Y = A, ad let B' be the compoet of f-l(b) which cotais f-l(b) Y B. It suffices to show that X = A' U B'. To this ed let x E K"Y, ad choose a sequece {x } covergig to h-l(x) such that x E C. Without loss of geerality ad by passig to sub sequeces we may assume that x E A for each. Let C' deote the compoet of UA which cotais x. Sice lim if{c'} ~ ~, it follows that lim sup{c'} ~ K is a co tiuum. Note that UA ~ A, for otherwise A would be a subcotiuum of U meetig distict compoets of U. Hece, c~(c~) bda(u A) ~ ~ where bda(m) deotes the boudary of M relative to A. It follows that (lim sup{c~}) K bd(u) A ~~. Sice lim sup{c~} ~ A, it follows that h(lim sup{c~}) is a subcotiuum of f-l(a) which meets A ad cotais x. Thus x E A' ad X = A' U B'. Theorem. If Y is a locally coected cotiuum which cotais a arc A such that each poit of A separates Y~ the there exist a cotiuum X ad a o-mootoe FD mappig f: X ~ Y. Proof. Let a ad b be the edpoits of A. Let A' be a arc with edpoits a' ad b', ad let X = Y U A' be the cotiuum obtaied by attachig a' to a ad b' to b. Let f: X ~ Y be a mappig so that f is the idetity o Y ad fla' is a homeomorphism of A' oto A. The f is clearly o-mootoe. To see that f is a FD mappig let Y = P U Q be a decompositio. By a result of Whybur [3, p. 51] all

5 16 Hughes but coutably may poits of A separate a from b i Y. It follows that if a ad b are both i P or Q, the A ~ P or A ~ Q, respectively. Otherwise, without loss of geerality, assume that a E P ad b E Q'P. we see that P A is coected. Agai usig Whybur's result I either case it is clear that there is a decompositio X = P' U Q' such that f(p') ~ P ad f (Q ') ~ Q. 3. Freely Decomposable Mappigs o Irreducible Cotiua The cotiuum X is said to be irreducible if there exist poits a ad b of X such that o proper subcotiuum of X cotais a ad b. Every FD mappig of a irreducible cotiuum oto a locally coected cotiuum is mootoe [1]. The followig theorem is a geeralizatio of that result. Theorem 3. If X is irreducible~ Y is semi-locally coected~ ad f: X + Y is a FD mappig~ the f is mootoe. Cosequetly~ if Y is odegeerate~ Y is a arc. Proof. Suppose f is ot mootoe. The there exist y E Y ad xl ad x i distict compoets of f-l(y). Let I C X be a cotiuum irreducible from xl to x ad let p E I'f-l(y). Because Y is semi-locally coected, there is a decompositio Y = A U B with f(p) E A\B ad y E B\A. Choose a decompositio X A' U B' such that f(a') c A ad f(b') ~ B. Let a E A' ad b E B' such that X is irreducible from a to b. Let J c B' be a cotiuum irreducible from xl to x. Choose q E J\(A' U f-l(y)). Let Y = CUD be a decompositio such that f(q) E C\D ad y E D\C, ad choose a decompositio X = C' U D' such that f(c') c C ad f(d') c D.

6 TOPOLOGY PROCEEDINGS Volume Case I: a E ct. Sice p B', there exists a ope set U such that p E U ~ ci(u) ~ A'\B'. For i = 1,, let Ii be the closure of the compoet of I\ci(U) which cotais Xi. The Ii ci(u) ~~. If q E II 1, the II U 1 is a proper subcotiuum of I cotaiig xl ad x which is a cotradictio. Thus assume without loss of geerality that q t II. It follows that A' U II U D' is a subcotiuur of X cotaiig a ad b but ot q. This is a cotradictio. Case II: a ED'. Let I' be the closure of the compoet of I\A' which cotais xl' ad let J' be the closure of the compoet of J\C' which cotais xl. The I' A' ~ ~ ~ J' ct. Sice p t I' ad q J', it follows that x t I' U J'. Thus A' U I' U J' U C' is a subcotiuur of X cotaiig a ad b but ot x This is a cotradictio. Sice f is mootoe, Y is irreducible. If Y is odegeerate, the Y is a arc by 6.3 of [4]. Refereces 1. G. R. Gordh, Jr. ad C. B. Hughes, O freely decomposable mappigs of cotiua, to appear i Glasik Matematicki.. F. B. Joes, Aposydetic cotiua ad certai boudary problems, Amer. J. Math. 63 (1941), G. T. Whybur, Aalytic Topology, AIDer. Math. Soc. Col loquium Publicatios 8, Providece, , Semi-locally coected sets, Amer. J. Math. 61 (1939), Uiversity of Ketucky Lexigto, Ketucky 40506