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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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##### References:
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