Homogeneity of Pixley-Roy spaces
Pixley-Roy hyperspaces were studied initially as sources of (counter)examples and later as objects of interest in their own right. This paper is an important contribution to our still meager understanding of the relationship between a topological space X and its Pixley-Roy hyperspace \({\mathcal F}[X].\) The author establishes the homogeneity of \({\mathcal F}[X]\) for a wide variety of spaces X. One class of such spaces includes the rationals, the irrationals, the Cantor set, the Alexandroff double arrow, the Sorgenfrey line, and all n-manifolds as well as finite products, open subspaces, and Pixley-Roy hyperspaces of members of the class. A larger class includes such non-homogeneous spaces as [0,1] and \({\mathbb{R}}^ 2/S^ 1\) and is closed under certain types of identification (identifying a finite subset to a point, for instance). All these spaces possess a property called base-chain homogeneity. The complexity of this concept and of the associated proofs prompts the author to ask whether there is an alternative representation of Pixley-Roy hyperspaces which makes them easier to study. A simpler, very different result is the following: Theorem 6. ``Let X be a \([T_ 1]\) space and assume that for all \(F,G\in {\mathcal F}[X]\) there is a homeomorphism from an open neighborhood of the point F in the identification space X/F to an open neighborhood of the point G in X/G that takes F to G. Then \({\mathcal F}[X]\) is homogeneous.'' While this theorem applies to only a few of the spaces listed above, it can be used for spaces of any character (whereas base-chain homogeneity implies first countability).
scientific article; zbMATH DE number 4071878
scientific article