A note on multivalued Baire category theorem. (English) Zbl 0904.54023

The author extends Baire’s result on iterations of continuous functions defined on complete metric spaces to the multivalued case. More specifically, the following theorem is obtained.
Let \(X_n\) be a sequence of regular topological spaces and let \(D_n\) be open dense subsets of respectively \(X_n\), \(n\geq 0\). Assume that one of the spaces \(X_n\) is a strongly countable Čech-complete space. If for every \(n>0\) the mapping \(F_n:X_n\to X_{n-1}\) is lower semi-continuous and its inverse is upper semi-continuous, then the set \(D=D_0\cap\bigcap_{n=1}^{\infty}F_1F_2\ldots F_n(D_n)\) is dense in \(X_0\).
Reviewer: D.Silin (Berkeley)


54E52 Baire category, Baire spaces
54C60 Set-valued maps in general topology
26E25 Set-valued functions
26A18 Iteration of real functions in one variable
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