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Confluent projections and connectedness of inverse limits. (English) Zbl 1476.54038

Inverse limits of continua, with single-valued mappings, are always connected. This is not the case when set-valued upper semi-continuous functions \(f_{n}:X_{n+1}\rightarrow 2^{X_{n+1}}\) are taken. Several authors have given conditions on the functions \(f_{n}\) in order that the inverse limit results connected. In this direction, V. Nall [Topol. Proc. 40, 167–177 (2012; Zbl 1261.54023)] considered set-valued functions \(f\) with the following properties:
(a)
the graph of \(f\) is compact and connected,
(b)
the graph of \(f\) is the union of a family of graphs of set-functions with connected images.
Then he proved that if \(X\) is the inverse limit taking the same set-valued function (\(f_{n}=f\) for every \(n\)), where \(f\) is a set-function satisfying (a) and (b), then \(X\) is connected.
In the paper under review the authors study set-functions satisfying condition (b) and other related conditions. For example, they prove that if \(f\) satisfies condition (b), then the projection from the graph of \(f\) onto the domain is confluent and that for arcwise connected continua the converse implication is also true.
The authors also show that Nall’s theorem also holds when each \(f_{n}\) is onto, satisfies (a) and (b) and they are not necessarily equal. They finish the paper by asking whether in this result it is possible to change the condition (b) by the assumption of the confluence of projections from the graph of \(f_{n}\) onto \(X_{n}\).

MSC:

54F17 Inverse limits of set-valued functions
54F15 Continua and generalizations
54C60 Set-valued maps in general topology
54D05 Connected and locally connected spaces (general aspects)
54E45 Compact (locally compact) metric spaces

Citations:

Zbl 1261.54023
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References:

[1] W. T. Ingram, An introduction to inverse limits with set-valued functions, Springer, 2012. · Zbl 1257.54033 · doi:10.1007/978-1-4614-4487-9
[2] W. T. Ingram and W. S. Mahavier, “Inverse limits of upper semi-continuous set valued functions”, Houston J. Math. 32:1 (2006), 119-130. · Zbl 1101.54015
[3] W. T. Ingram and W. S. Mahavier, Inverse limits, Developments in Mathematics 25, Springer, 2012. · Zbl 1234.54002 · doi:10.1007/978-1-4614-1797-2
[4] V. Nall, “Connected inverse limits with a set-valued function”, Topology Proc. 40 (2012), 167-177 · Zbl 1261.54023 · doi:10.1109/tps.2011.2171372
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