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

Urbaniec, Piotr

Acta Univ. Carol., Math. Phys. 37, No. 1, 11-15 (1996).

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)

MSC:

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

Keywords:

Baire category; set-valued function; dense image