| An Uncountable Collection of Mutually Incomparable Planar Fans |
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| Escrito por Carlos Islas |
| Miércoles, 17 de Noviembre de 2010 11:19 |
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Abstract. In this paper, we show an uncountable collection of mutually incomparable fans in the plane. 1. Introduction A continuum means a nonempty compact connected metric space and a map means a continuous function. An arc means a space homeomorphic to the closed unit interval [0, 1]. A continuum X is arcwise connected provided each two points of X are contained in some arc contained in X. A continuum X is unicoherent if A∩B is connected for each subcontinua A and B such that A ∪ B = X. X is hereditarily unicoherent if every subcontinuum of X is unicoherent. A dendroid is an arcwise connected hereditarily unicoherent continuum. A fan is a dendroid with only one ramification point. We say that two continua are comparable by continuous maps if one of those continua can be mapped onto the other. Otherwise, the continua are incomparable. In 1961, B. Knaster [2] asked for an uncountable family of continuously incomparable dendroids. Recently, this question was answered independently in [3] by Piotr Minc and in [4]. These dendroids are fans, but it is not clear if they are planable. Minc presented his example during the Spring Topology Conference held in Greensboro, North Carolina, in March 2006. |
