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On generalized Whitney mappings. (English) Zbl 0551.54007
Let \(X\) be a compact space and \(H(X)\) its hyperspace. A map \(w: H(X)\to [0,1]\) is called a Whitney map if it is a continuous monotonic function such that \(w(\{x\})=0\) for each \(x\leq X\), and such that \(w(A)<w(B)\) whenever A is properly contained in B. The author introduces somewhat weaker conditions than Whitney maps do and gives equivalent conditions for the existence of such functions. Let \(w: H(X)\to [0,1]\) be a continuous monotonic function with \(w(\{x\})=0\) for each \(x\in X\) and consider the following properties for w: Whitney-like: \(w(A)\leq w(B)\) whenever \(A\subset B\) and \(int_ X(A\setminus B)\neq \phi\), (I) \(w(A)<w(X)\) for each proper closed subset \(A\in H(X)\), (II) \(w(A)<w(X)\) for each nowhere dense set \(A\in H(X)\). It is clear that Whitney \(\Rightarrow\) Whitney-like \(\Rightarrow\) (I) \(\Rightarrow\) (II).
The author proves the following theorems: (1.2.) Let \(X\) be a compact Hausdorff space. Then the following assertions are equivalent: (i) \(X\) has a countable \(\pi\)-weight; (ii) X has a Whitney-like map; (iii) there is a map \(w : H(X)\to [0,1]\) with property (I). (1.3.) Let \(X\) be a compact Hausdorff space. Then the following conditions are equivalent: (1) there is a countable collection \(\{0_ n:n\in N\}\) of non-empty open sets such that each dense open set of \(X\) contains some \(0_ n\); (2) there is a map \(w: H(X)\to [0,1]\) with property (II). The author also investigates the relationship between these functions and Whitney maps on certain quotients.
Reviewer: A.Okuyama
MSC:
54B20 Hyperspaces in general topology
54C10 Special maps on topological spaces (open, closed, perfect, etc.)
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