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Inverse limits of upper semi-continuous set valued functions. (English) Zbl 1101.54015

Inverse limits are usually defined for sequences \((X_1,f_1), (X_2,f_2), (X_3,f_3), \cdots\) such that for each \(i\), \(X_i\) is a topological space and \(f_i\) is a map from \(X_{i+1}\) to \(X_i\). In this paper, the authors consider sequences \((X_1,f_1), (X_2,f_2), (X_3,f_3), \cdots\) such that for each \(i\), \(X_i\) is a compact Hausdorff space and \(f_i\) is an upper semi-continuous function from \(X_{i+1}\) to \(2^{X_i}\). Such a sequence is called an inverse limit sequence and the functions \(f_i\) are called bonding functions. The inverse limit is defined to be the set of all points \((x_1, x_2, x_3,\cdots)\in\prod X_i\) such that \(x_i\in f_i(x_{i+1})\) for each \(i\), and is denoted by \(\lim_{\leftarrow}\mathbf{f}\), where \(\mathbf{f}=f_1, f_2, f_3,\cdots\). They study properties of inverse limit sequences of compact Hausdorff spaces \(X_i\) with upper semi-continuous bonding functions \(f_i:X_{i+1}\to 2^{X_i}\).
Some of the main results are: (1) The limit is non-empty and compact. (2) Assume that for each \(i\), \(X_i\) is a Hausdorff continuum and for each \(x\in X_{i+1}\), \(f_i(x)\) is connected. Then the limit is a Hausdorff continuum. (3) Suppose \((X_1,f_1), (X_2,f_2), (X_3,f_3), \cdots\) and \((Y_1,g_1), (Y_2,g_2), (Y_3,g_3), \cdots\) are inverse limit sequences of compact Hausdorff spaces with upper semi-continuous bonding functions. Suppose further that, for each \(i\), \(\varphi_i:X_i\to Y_i\) is a map such that \(\varphi_i\circ f_i=g_i\circ\varphi_{i+1}\). Then the function \(\varphi:\lim_{\leftarrow}\mathbf{f}\to\lim_{\leftarrow}\mathbf{g}\) defined by \(\varphi(\mathbf{x})=(\varphi_1(x_1), \varphi_2(x_2), \varphi_3(x_3), \cdots)\) for \(\mathbf{x}=(x_1, x_2, x_3, \cdots)\in\lim_{\leftarrow}\mathbf{f}\) is continuous. Furthermore, if each \(\varphi_i\) is injective, so is \(\varphi\), and if each \(\varphi_i\) is surjective, so is \(\varphi\). Some interesting examples in the special case where each \(X_i\) is the closed unit interval are given.

MSC:

54B35 Spectra in general topology
54C60 Set-valued maps in general topology
54B10 Product spaces in general topology
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