Whitney Maps - a Non - Metric Case
نویسنده
چکیده
It is shown that there is no Whitney map on the hyperspace 2x for nonmetrizable Hausdorff compact spaces X. Examples are presented of non-metrizable continua X which admit and ones which do not admit a Whitney map for C(X). Given a Hausdorff compact space X, we consider the space 2x of all non-empty compact subsets of X equipped with the Vietoris topology. Any subspace H (X) of the space 2x is called a hyperspace of X. In particular Fn(X) stands for the family of all non-empty subsets of X of cardinality at most n (where n EN), and C(X) denotes the hyperspace of sub continua of X (i.e., of connected members of 2X). The reader is referred to [4] and [5] for needed information on hyperspaces. A continuum X containing two points a and b is called an arc (from a to b) provided that each point of X \ {a, b} separates a and b in X. We write ab to denote an arc with end points a and b. Note that an arc is metrizable if and only if it is homeomorphic to the closed unit interval [0,1]. Given a Hausdorff compact space X and its hyperspace H(X), by a Whitney map for H(X) we mean a mapping J1 : H(X) -+ ab such that (0.1) J1({x}) = a for each point x E X; (0.2) A <; B implies J1(A) < J1(B). When X is a compact metric space, then a Whitney map for 2x or C(X) does always exist, and several constructions of such mappings are known: see e.g. [5, 0.50.1-0.50.3, pp. 25-26] or [4, Theorem 13.4, p. 107; Exercises 13.5-13.8, pp. 108-109]. The following theorem answers a question of Robert Heath asked during a private conversation with the second named author. THEOREM 1. The following conditions are equivalent for a Hausdorff compact space X: 2000 Mathematics Subject Classification: 54B20, 54C99, 54E35, 54F15.
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