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On Bellamy’s set function \(\Gamma\). (English) Zbl 1469.54022

A compactum is a compact connected metric space. A continuum is a connected compactum. For a compactum \(X\), let \(2^X\) be the hyperspace of all nonempty closed subsets of \(X\) and let \(C(X)\) be the hyperspace of all subcontinua of \(X\). Both hyperspaces are topologized with the Vietoris topology.
Given a compactum \(X\), we define Jones’ set function \(\mathcal T\) as follows: if \(A\) is a subset of \(X\), then \[\mathcal T(A) = X \setminus \{ x \in X : \exists W \in C(X) \ (x \in \text{Int}_X W \subseteq W \subseteq X\setminus A)\};\] and we define Bellamy’s set function \(\Gamma\) as follows: if \(A\) is a subset of \(X\), then \[\Gamma(A) = X \setminus \left\{ x \in X : \begin{array}{c} \exists L \in 2^X \ (x \in \text{Int}_X L \subseteq L \subseteq X \setminus A \ \text{and} \\ L \ \text{has at most countably many components}) \end{array}\right\}\]
The aim of this paper is to investigate Bellamy’s set function \(\Gamma\). Many of the results presented are “parallel” to ones known for Jones’ set function \(\mathcal T\).
The first part is dedicated to giving general properties of the set function \(\Gamma\). Some of such properties are: the condition \(\Gamma(\emptyset) = \emptyset\) characterizes compacta having at most countably many components; the condition \(\Gamma(A)\) is totally disconnected for a nonempty closed subset \(A\) of a continuum implies that \(\Gamma(A) = A\); all continua such that the restriction of \(\Gamma\) to \(2^X\) is the identity map for \(2^X\) are characterized; each continuum such that the image of a one-point set under \(\Gamma\) is the whole continuum must be indecomposable; if \(f\) is a monotone and open mapping from a continuum \(X\) onto a continuum \(Y\), then \(f^{-1}(\Gamma_Y(B)) = \Gamma_X(f^{-1}(B))\) for each nonempty subset \(B\) of \(Y\); and finally, the image of a one-point subset of a metric homogeneous continuum under \(\Gamma\) and its image under \(\mathcal T\) coincide.
The second part is devoted to a study of the symmetry, additivity and idempotency of \(\Gamma\). Some of such results are: the symmetry of \(\mathcal T\) implies the symmetry of \(\Gamma\); the additivity of \(\Gamma\) is characterized in terms of families of closed subsets of the continuum \(X\) whose union is closed in \(X\); and the set function \(\Gamma\) is not idempotent on the product of a continuum and an indecomposable continuum.
Finally, the continuity of the restriction of \(\Gamma\) to \(2^X\) is characterized and the following fact is proved: the collection of subsets \(\{ \Gamma(\{x\}) : x \in X\}\) is a continuous decomposition provided that \(\Gamma\) is idempotent on singletons and continuous on singletons.

MSC:

54F16 Hyperspaces of continua
54B20 Hyperspaces in general topology
54C60 Set-valued maps in general topology
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References:

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[16] Instituto de Matemáticas;
[17] Circuito Exterior, Ciudad Universitaria;
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