Quasi-monotone mappings on θn-continua |
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Authors: | E.E. Grace Eldon J. Vought |
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Affiliation: | Department of Mathematics, Arizona State University, Tempe, AZ 85287, USA;Department of Mathematics, Chico State University, Chico, CA 95929, USA |
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Abstract: | A continuous function f from a continuum X onto a continuum Y is quasi-monotone if, for every subcontinuum M of Y with nonvoid interior, f-1(M) has a finite number of components each of which is mapped onto M by f. A θn-continuum is one that no subcontinuum separates into more than n components. It is known that if f is quasi-monotone and X is a θ1-continuum, then Y is a θ1-continuum or a θ2-continuum that is irreducible between two points. Examples are given to show that this cannot be generalized to a θn-continuum and n + 1 points for any n >1, but it is proved that if f is quasi-monotone and X is a θn-continuum, then Y is a θn-continuum or a θn+1-continuum that is the union of n + 2 continua H,S1,S2,…,Sn+1, whe for each i, Si is the closure of a component of Y H, Si is irreducible from some point Pi to H, and H is irreducible about its boundary. Some theorems and examples are given concerning the preservation of decomposition elements by a quasi-monotone map defined on a θn-continuum that admits a monotone, upper-semicontinuous decomposition onto a finite graph. |
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Keywords: | 54C10 54F20 54B15 54E25 54G20 quasi-monotone mapping ω-connected condensation decomposition δ-connected set of irreducibility indecomposable continuum |
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