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Confluent set-valued functions and inverse limits. (English) Zbl 1440.54023

For a metric continuum \(Y\) let \(2^{Y}\) denote the hyperspace of all nonempty closed subsets of \(Y\). The function \(f: X \to 2^{Y}\) from a continuum \(X\) to the hyperspace \(2^{Y}\) is called set-valued function. The set \(G(f) = \{(x,y) \in X \times Y : y \in f(x) \}\) is the graph of \(f\).
In [Topology Appl. 228, 486–500 (2017; Zbl 1378.54033)] J. P. Kelly introduced a notion of monotone and confluent set-valued function: the upper semi-continuous function \(f: X \to 2^{Y}\) is monotone (confluent) if the projections \(\pi _{X} : G(f) \to X\) and \(\pi _{Y} : G(f) \to Y\) are monotone (confluent). Considering inverse limits with set-valued functions, J. P. Kelly shows that if each factor space is an arc and the bonding functions are monotone set-valued functions, then the projection maps from the inverse limit and from the inverse graph are also monotone. He asks if analogous results would be true in the case of confluent set-valued functions. In the paper under review the authors give a positive answer to this question.
In [Commentat. Math. Univ. Carol. 23, 183–191 (1982; Zbl 0486.54029)] W. J. Charatonik proved that an inverse limit, whose factor spaces are continua with the property of Kelley and the bonding (single-valued) maps are confluent has the property of Kelley. In the above-mentioned paper Kelly asks if this result is still true in the case of inverse limits with set-valued functions. The authors give a negative answer to this question.

MSC:

54F17 Inverse limits of set-valued functions
54F15 Continua and generalizations
54C60 Set-valued maps in general topology
54C05 Continuous maps
54C10 Special maps on topological spaces (open, closed, perfect, etc.)
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References:

[1] Charatonik, W. J., Inverse limits of smooth continua, Comment. Math. Univ. Carol., 23, 1, 183-191 (1982) · Zbl 0486.54029
[2] Ingram, W. T., An Introduction to Inverse Limits with Set-Valued Functions, Springer Briefs in Mathematics (2012), Springer: Springer New York · Zbl 1257.54033
[3] Ingram, W. T., Inverse limits of families of set-valued functions, Bol. Soc. Mat. Mex. (3), 21, 1, 53-70 (2015) · Zbl 1319.54008
[4] Kelley, J. L., Hyperspaces of a continuum, Trans. Am. Math. Soc., 52, 22-36 (1942) · Zbl 0061.40107
[5] Kelly, J. P., Monotone and weakly confluent set-valued functions and their inverse limit, Topol. Appl., 228, 486-500 (2017) · Zbl 1378.54033
[6] Macías, S., Topics on Continua (2018), Springer: Springer Cham, previously published by Chapman & Hall/CRC 2005 · Zbl 1403.54001
[7] Wardle, R. W., On a property of J.L. Kelley, Houst. J. Math., 3, 291-299 (1977) · Zbl 0355.54025
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