Images of the Cantor fan
A continuum \((=\) compact, connected, metric space) is hereditarily unicoherent if the intersection of any two subcontinua is connected. If it is also arcwise connected it is called a dendroid. If a dendroid has exactly one ramification point v, it is called a fan. The cone over the Cantor set is a fan which is called the Cantor fan \(F_ c\). The author is mainly concerned with characterizing the image of \(F_ c\) under confluent, open, monotone, retraction or light mappings. Five characterizations of the image of \(F_ c\) under a confluent mapping are given. For example, the image of \(F_ c\) is either an arc or a fan having the property of Kelley (a property similar to smoothness). Or, the image of \(F_ c\) is either an arc or a smooth fan whose set of end points union its ramification point v is closed. Similarly, six characterizations of the image of \(F_ c\) under an open mapping are given. For example, this image is either an arc or a smooth fan whose set of end points is closed. For monotone mappings there are two characterizations, one of which says: The nondegenerate continuum Y is the image of \(F_ c\) under a monotone mapping if and only if F is homomorphic either to \(F_ c\) or to \(F^ w_ c\). Finally, the author shows that the image of \(F_ c\) under a mapping (continuous, of course) is the image of \(F_ c\) under a light mapping.
scientific article; zbMATH DE number 4143014